Discussion Paper No. 706

DETERMINACY OF EQUILIBRIUM
UNDER VARIOUS PHILLIPS CURVES

Yoshiyasu Ono

February 2008

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

February 2008

Determinacy of Equilibrium under Various Phillips Curvesby

Yoshiyasu Ono*
Osaka University

Abstract
Determinacy of equilibrium under the original, the backward-looking, theforward-looking and the hybrid Phillips curves is examined. If the monetary authority keeps
the nominal money stock to be constant, the equilibrium path is always determinate under theoriginal Phillips curve and the forward-looking one. Under the backward-looking one and the
hybrid one, however, the path can be non-existent. The case of a Taylor rule is also examined.
Under any of the four curves the path is always determinate if the monetary policy is active but
is never determinate if it is passive.
Keywords: Phillips curve, indeterminacy, non-existence, Taylor rule.
JEL Classification Numbers: E52, E31
Address: Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka,
Ibaraki, Osaka, 567-0047, JAPANE-mail: ono@iser.osaka-u.ac.jp*

I would like to thank Y. Fukuta, S. Ikeda, A. Kaneko, R. Murota, F. Ohtake, M. Saito and Y. Tsutsui for theirhelpful comments and suggestions. I have particularly benefited from discussions with A. Shibata. Comments by
the seminar participants at Tokyo Institute of Technology and Kyoto University are gratefully acknowledged.
This research is financially supported by the Grants-in-Aid for Scientific Research, JSPS.

1. Introduction
This paper examines the existence of a unique equilibrium path under various Phillips
curves, viz. the original Phillips curve, the backward-looking one, the forward-looking one andthe hybrid of the backward- and forward-looking ones. It does so under a money stock control
rule and interest-rate feedback rules (Taylor rules) and finds that the path is non-existent orindeterminate under various situations.
Since Phillips (1958) found a negative relationship between the inflation rate and theunemployment rate, various attempts to elaborate the Phillips curve have been made. Friedman(1968) and Phelps (1967) introduce the expected inflation rate into the original Phillips curveand present a backward-looking Phillips curve, called the New Classical Phillips Curve.1 Morerecently microeconomic foundations of nominal price adjustment have been analyzed and aforward-looking Phillips curve, called the New Keynesian Phillips Curve, is proposed insteadof the backward-looking one. Roberts (1995) derives it from the models of staggered contracts
developed by Taylor (1979, 1980), Rotemberg (1982) and Calvo (1983).2
However, empirical fitting of the New Keynesian Phillips Curve is found to be
unsuccessful. Thus, a hybrid type of the backward- and forward-looking Phillips curves is
proposed and shown to fit data better than the others. Fuhrer and Moore (1995), Gali and
Gartler (1999), Christiano et al. (2005), Rudd and Whelan (2005a, 2005b), Sawyer (2007),
Smith and Wickens (2007) are such examples.
Apart from those empirical analyses Benhabib et al. (2001a) theoretically examine thestability of macroeconomic dynamics under the forward-looking Phillips curve when themonetary authority follows a Taylor rule. They find that the equilibrium path is determinate if
the authority moves the nominal interest rate so much that the real interest rate also moves inthe same direction as the inflation rate does (viz. active feedback) but indeterminate if the
feedback is not so much (viz. passive feedback). Benhabib et al. (2001b) analyze the stability
in the case of multiple steady states and find that the equilibrium path is indeterminate evenunder the active feedback. Woodford (2003) extensively studies this issue under the1

This type of Phillips curve is adopted by Kydland and Prescott (1977) and Barro and Gordon (1983) whenanalyzing the effects of monetary policies.
2

Kimball (1995) and Yun (1996) incorporate Calvo’s price-setting model into a dynamic general equilibrium
framework.

1forward-looking Phillips curve and various Taylor rules and shows that the path can be
determinate or indeterminate, depending on the parameter values of the Taylor rules.3
Since they focus on the dynamic performance of Taylor rules rather than that of thePhillips curve, they treat only the cases of flexible prices and the forward-looking Phillips
curve:4 the cases of the other Phillips curves are not examined. In contrast, this paper focuseson the dynamic stability under various Phillips curves, viz. the original, the backward-looking,the forward-looking and the hybrid ones, and explores under which type of Phillips curve theequilibrium path is determinate. Furthermore, since a money stock control rule instead of a
Taylor rule is mostly assumed in the monetary-growth literature, it begins with the case of a
fixed amount of nominal money stock. It is found that there is a case where the path isnon-existent under the backward-looking Phillips curve and the hybrid one whereas it isalways determinate under the original Phillips curve and the forward-looking one.
Determinacy of equilibrium under a Taylor rule is also examined in each case. It is shown
that, as long as the Taylor rule is passive, the equilibrium path is never determinate under anyof the four Phillips curves. It is either non-existent or indeterminate. If it is active, theequilibrium path is always determinate under any of them. Thus, if a Taylor rule is adopted, themonetary policy must be active in order for the equilibrium path to be determinate.

2. The Model
The household sector receives income y, which equals actual total production, and holds
real balances m. For simplicity, it is assumed that m is the only storable asset. The household
sector determines c and m so as to maximize lifetime utility U:∞

U = ∫ u (c, m)e −ρt dt ,0

subject to the flow budget equation:
m& = y − c − πm + τ,(1)

where ρ is the subjective discount rate, π is the inflation rate of commodity price P, y is3

The possibility of equilibrium indeterminacy under a Taylor rule is also explored in the presence of capital
accumulation. See Itaya and Mino (2004) and Meng and Yip (2004) for the case of flexible prices and Dupor(2001) and Carlstrom and Fuerst (2005) for the case of the forward-looking Phillips curve in a continuous-timeand discrete-time setting respectively.
4

Schmitt-Grohé and Uribe (2000) analyze the dynamic stability under a Taylor rule and flexible prices andfind that there are the case with a continuum of equilibrium paths and that with a unique equilibrium path.

2

aggregate demand, and τ is the government’s lump-sum transfer.
Naturally, u(c, m) is assumed to satisfy the following properties:

Assumption 1.

uc > 0, um > 0, ucc < 0, umm < 0, Φ = ummucc − ucm2 > 0.
∂(um/uc)/∂c = (umcuc − uccum)/uc2 > 0,

Assumption 2.

∂(um/uc)/∂m = (ummuc − ucmum)/uc2 < 0,

1 + ucmm/uc > 0.

Assumption 1 presents the standard conditions, including concavity, that a utility function mustsatisfy. The first two properties of assumption 2 imply normality of consumption c and of realbalances m, respectively. The third property means that ucm is not significantly negative. It isdefinitely valid either if c and m are Edgeworth substitutes (viz. ucm > 0) or if u(c, m) is additive
separable (viz. ucm = 0).
From the Hamiltonian function of this problem:
H = u(c, m) + λ(y − c − πm + τ),
one derives the first-order optimal conditions that are summarized as follows:R=u m (c, m)u (c, m)c& + u cm (c, m)m&= ρ + π − cc,u c (c , m )u c ( c, m )(2)where a dot represents a time derivative, each subscript implies the partial derivative withrespect to it, and R is the nominal interest rate. The transversality condition islim λ t mt exp(−ρt ) = 0

t →∞where λt = uc(ct, mt).

(3)

The equilibrium of the money market is given by
M/P = m,(4)

where M is the nominal money stock. If commodity price P is flexible, the present model is thesame as that of Obstfeld and Rogoff (1983) and thus the unique equilibrium path obtains.5
Alternatively, this paper assumes P to be sluggish and considers the following four types ofcommodity price adjustment:

5

Original: π = f1(y/yn),(5a)Backward-looking: π& = f2(y/yn),(5b)

The path is such that P initially jumps to the steady-state level and stays there.

3Forward-looking: π& = ρπ − f3(y/yn),(5c)Hybrid: &π& = ρ π& − f4(y/yn),(5d)where fi(1) = 0 and fi′ > 0 for i = 1, 2, 3, 4.
In the above equations yn is the natural rate of output and ρ is the subjective discount rate.
(5a) is clearly the original Phillips curve. Note that y and π have a one-to-one
correspondence and hence the degree of freedom is 1 when choosing the initial levels of thetwo variables. (5b) is the continuous version of the backward-looking adjustment (viz. theNeoclassical Phillips Curve) proposed by Phelps (1967) and Friedman (1968):
πt = πt−1 + γln(yt /yn).

It reduces toπt − πt−1 = γln(yt /yn),which is essentially the same as (5b). Since πt−1 is predetermined, πt is determined once yt isdetermined –i.e., the degree of freedom is 1 when choosing the initial levels of y and π.
(5c) is the continuous version of the forward-looking adjustment (viz. the New KeynesianPhillips Curve) presented by Roberts (1995):
πt = βEt[πt+1] + κln(yt /yn),where β is the subjective discount factor. This equation reduces toπt+1 − πt = [(1 − β)/β]πt − (κ/β)ln(yt /yn),which is essentially the same as (5c). Note that discount factor β equals 1/(1 + ρ) and hence thecoefficient of πt in the right-hand side equals ρ, as presented in (5c). Since there is πt+1 in thisequation, yt and πt are both jump variables.
The hybrid model by Christiano et al. (2005) (see also Woodford, 2003, p.215) is
πt − πt−1 = βEt[πt+1 − πt] + κln(yt /yn),which reduces to(πt+1 − πt) − (πt − πt−1) = [(1 − β)/β](πt − πt−1) − (κ/β)ln(yt /yn).
(5d) is its continuous version. Since πt−1 is predetermined, πt and yt are jump variables.
However, once they are chosen, πt − πt−1 and πt+1 − πt are determined. Thus, the degree of
freedom is 2 when choosing the initial levels of y, π and π& in this case.
Needless to say, each fi(y/yn) (for i = 1, 2, 3, 4) is a generalized form of γln(y/yn) or(κ/β)ln(y/yn) discussed above.

43. Money Stock Control Rule
This section assumes a simple money stock control rule, viz. keeping money stock M to beconstant. Then, from (4),m& = − πm.

(6)Fiscal spending is assumed to be zero and thus transfer τ is zero. In this case (1) and (6) imply
y = c.

(7)

Applying (6) and (7) to (2) gives the dynamic equation of y:⎛ u ( y , m) ⎞ ⎡ u m ( y , m)
⎛ u ( y , m) m ⎞ ⎤⎟⎟π⎥ .
⎟⎟ ⎢− ρ − ⎜⎜1 + cm
y& = ⎜⎜ − c
u
(
y
,
m
)
u
(
y
,
m
)
u
(
y
,
m
)
c
⎝ cc
⎠⎣ c
⎝
⎠ ⎦

(8)

Equations (6), (8) and one of the four equations in (5) formulate an autonomous dynamic
system. Note that the four scenarios have the same steady state represented by(π, y, m) = (0, yn, m*), whereu m ( y n , m*)= ρ.
u c ( y n , m*)(9)

In this state transversality condition (3) is obviously valid.6

3.1. Original Phillips curve

Substituting π given by (5a) into (6) and (8) yields⎤⎛ u ( y , m) m ⎞⎛ u ( y , m) ⎞ ⎡ u m ( y , m)⎟⎟ f 1 ( y / y n )⎥ ,⎟⎟ ⎢− ρ − ⎜⎜1 + cm
y& = ⎜⎜ − cu c ( y , m) ⎠⎝ u cc ( y, m) ⎠ ⎣⎢ u c ( y, m)⎝⎦⎥
m& = − f1(y/yn)m.

Therefore, around the steady state given by (9) the characteristic equation of the (y, m)dynamics isy& y − λ y& m
f ′m *
= 0,
− 1n
−λ
y
where under assumption 2 y& m satisfies

6

There may be another steady state in which m expands to infinity, um = 0, and y takes the value that makes y&given by (8) equal zero. In the case where ucm = 0, for example, such a state indeed obtains when ρ + π = 0 and ytakes the value that makes π equal − ρ under each Phillips curve (5a) – (5d). However, from (2), λ in (3) and (6),λ& / λ + m& / m − ρ = um/uc = 0 in this state, implying that transverslity condition (3) is invalid. Therefore, the paththat approaches this state cannot be an equilibrium path.

5⎛ u ⎞ ∂ (u m / u c )y& m = ⎜⎜ − c ⎟⎟< 0.
∂m
⎝ u cc ⎠(10)

The characteristic function reduces to
λ2 + Aλ + B = 0,
where using (10) one finds

B = λ1λ2 = f1′ y& m m*/yn < 0,and λ1 and λ2 are the two characteristic roots. Therefore, the two roots are both real numbers,
among which one is positive and the other is negative. Since y is jumpable while m is not, theequilibrium path is uniquely determined.

3.2. Backward-looking adjustment

Around the steady state given by (9) the characteristic function of the (π, y, m) dynamics
formulated by (5b), (6) and (8) is−λy& π

f 2′ / y n
y& y − λ

− m*

0

0
y& m = 0,

(11)

−λ

where y& m is the same as given by (10). Under assumptions 1 and 2 y& π and y& y satisfy⎛ u ⎞ ⎛ u m⎞y& π = ⎜⎜ c ⎟⎟ ⎜⎜1 + cm ⎟⎟ < 0,uc ⎠⎝ u cc ⎠ ⎝⎛ u ⎞ ∂ (u m / u c )y& y = ⎜⎜ − c ⎟⎟> 0.
∂c
⎝ u cc ⎠(12)

The characteristic function (11) reduces to
λ3 + Aλ2 + Bλ + D = 0,
where using (10) and (12) one finds
A = − (λ1 + λ2 + λ3) = − y& y < 0,B = λ1λ2 + λ2λ3 + λ3λ1 = − f2′ y& π /yn > 0,
D = − λ1λ2λ3 = f2′ y& m m*/yn < 0.

(13)First, suppose that all the roots are real numbers. Since neither π nor m is jumpable
whereas y is jumpable, in order that there is a unique equilibrium path one of the roots is
positive and the other two are negative. Calling the positive root λ1 and the two negative roots
λ2 and λ3, the first property of (13) gives6

0 < − (λ2 + λ3) < λ1,which implies
B = λ1(λ2 + λ3) + λ2λ3 < λ2λ3 − (λ2 + λ3)2 < 0.

This property contradicts the second property of (13), and hence the case of one positive andtwo negative roots never occurs. From this result and the third property of (13), all roots are
positive, implying that there is no equilibrium path.
If two of the three roots are complex numbers, which must be conjugate, the third propertyof (13) implies the real root to be positive. From the first and second properties the real part ofthe complex roots can be either positive or negative. If it is positive, all paths are unstable andhence there is no equilibrium path. If it is negative, the equilibrium path is determinate. At theend of this section it is shown that the two cases in fact arise.

3.3. Forward-looking adjustment

Around the steady state given by (9) the characteristic function of the (π, y, m) dynamics
formulated by (5c), (6) and (8) isρ − λ − f 3′ / y n
y& π
y& y − λ0y& m = 0.

− m*

−λ

0

This equation reduces to
λ3 + Aλ2 + Bλ + D = 0,
where from (10) and (12)A = − (λ1 + λ2 + λ3) = − (ρ + y& y ) < 0,
D = − λ1λ2λ3 = − f3′ y& m m*/yn > 0.

(14)

If all the roots are real numbers, from the second property of (14) λi’s (i = 1, 2, 3) satisfy
either λ1 > 0, λ2 > 0, λ3 < 0;or λ1 < 0, λ2 < 0, λ3 < 0.
Since the second case contradicts the first property of (14), only the first case is valid, implyingthat there are two positive roots and a negative one. Since π and y are jump variables whereas m
is not, there is a unique equilibrium path in this case.
If two of the three roots are complex numbers, which must be conjugate, from the second
property of (14) the real root is negative and from the first property of (14) the common real7

part of the complex roots is positive. Therefore, the equilibrium path is determinate whether or
not all roots are real numbers.

3.4. Hybrid adjustment

Since (5d) is a second-order differential equation, it is decomposed to the following two
equations:
π& = z,z& = ρz − f4(y/yn).

(15)

The (π, z, y, m) dynamics is given by (6), (8) and the two equations in (15). Around the steadystate given by (9) the characteristic function is
−λ

1

0

ρ−λ

y& π
− m*

0

0
f′
− 4n
y
y& y − λ

0

y& m

0

0

−λ

0

= 0,

where y& π , y& y and y& m are given by (10) and (12). It reduces to
λ4 + Aλ3 + Bλ2 + Dλ + E = 0,
where using (10) and (12) one finds
A = − (λ1 + λ2 + λ3 + λ4) = − (ρ + y& y ) < 0,
B = (λ1 + λ3)(λ2 + λ4) + λ1λ3 + λ2λ4 = ρ y& y > 0,
D = − (λ1λ2λ3 + λ2λ3λ4 + λ3λ4λ1 + λ4λ1λ2) = f4′ y& π /yn < 0,E = λ1λ2λ3λ4 = − f4′ y& m m*/yn > 0.

(16)

First, suppose that all roots are real numbers. Since m is not jumpable and only two of y, π
and z are jumpable, as discussed at the end of section 2, two of the four roots (called λ1 and λ2)
are positive and the other two are negative (called λ3 and λ4) in order that the equilibrium pathis determinate. In this case λ1λ3 + λ2λ4 < 0 and hence from the first and the second properties of(16) one finds
λ1 + λ3 > 0, λ2 + λ4 > 0.
Since both λ1λ3 and λ2λ4 are negative, the above property yields
λ1λ3(λ2 + λ4) + λ2λ4(λ1 + λ3) < 0,8

which contradicts the third property of (16). Therefore, the case of two positive and two
negative roots never occurs. Moreover, from the first and last properties of (16) it follows that
all roots are positive, implying that there is no equilibrium path.
Next, when two roots (λ1 and λ2 without loss of generality) are real numbers and the othertwo are mutually conjugate complex numbers (δ ± εi), (16) reduces to− A = λ1 + λ2 + 2δ > 0,
B = λ1λ2 + 2δ(λ1 + λ2) + δ2 + ε2 > 0,

− D = 2δλ1λ2 + (λ1 + λ2)(δ2 + ε2) > 0,E = λ1λ2(δ2 + ε2) > 0.

(17)

From the fourth property of (17), λ1 and λ2 are either both negative or both positive. If theformer is right, the first property implies
δ > − (λ1 + λ2)/2 > 0.
From this property one obtains
2δλ1λ2 + (λ1 + λ2)(δ2 + ε2) < δ[2λ1λ2 + (λ1 + λ2)δ]< δ[2λ1λ2 − (λ1 + λ2)2/2] = − δ(λ1 − λ2)2/2 < 0,which contradicts the third property of (17) and hence both λ1 and λ2 are positive. The value ofδ that is consistent with all the four properties can be either positive or negative. The
equilibrium path is non-existent if δ is positive, and determinate if it is negative. It is latershown that the two cases indeed occur.
Finally, suppose that all roots are conjugate complex numbers:
δ ± εi and θ ± νi.
Then, the first and second properties of (16) reduce to(ρ + y& y )/2 = δ + θ,
ρ y& y = (θ + δ)2 + ν2 + 2δθ + ε2.
Since they yield(ρ − y& y )2/4 + ν2 + 2δθ + ε2 = 0,
δ and θ must have mutually different signs –i.e., the equilibrium path is determinate.
The results of this section are summarized as follows:

Proposition 1. Suppose that the monetary authority keeps the nominal money stock to beconstant. Under the backward-looking Phillips curve and the hybrid one the equilibrium9path is either non-existent or determinate. Under the forward-looking Phillips curve andthe original Phillips curve the equilibrium path is always determinate.

As mentioned in Proposition 1, the equilibrium path may be non-existent under the
backward-looking Phillips curve and the hybrid one. Using an additive separable utility
function in c and m (viz. ucm = 0) one can explicitly obtain the condition with respect to ηm, ηc
and fi′ (i = 2 or 4) for the path to be non-existent (or determinate) under the backward-looking
Phillips curve and the hybrid one, where ηm = − ummm/um, the elasticity of the marginal utilityof money, and ηc = − uccc/uc, the elasticity of the marginal utility of consumption. It is formallystated in proposition 2, of which the proof is set out in the appendix.

Proposition 2. Suppose that the monetary authority keeps the nominal money stock to beconstant and that the utility function is additive separable. Under the backward-looking
Phillips curve the equilibrium path is non-existent if and only if ηm < 1. Under the hybrid
Phillips curve the path is non-existent if and only if ηm < 1/2 − (f4′/ηc)/(4ρ3) (see figure 1).7

Since nominal interest rate R satisfies (2), if the utility function is additive separable, one
finds(m/R)∂R/∂m = ηm.
If R has a positive lower bound R0, as is the case under a liquidity trap, ηm approaches zero as mincreases.8 Thus, ηm is very small if m is large enough in the steady state. Proposition 2 implies
that if the equilibrium path is determinate in this case, the Phillips curve is neither the
backward-looking type nor the hybrid type.

4. Taylor Rule
This section examines the dynamic stability under a Taylor rule. Nominal money stock M
is controlled so that nominal interest rate R given by (2) should be a function of π:7Note that f4′ has the same order as ρ3 and &π& , as is clear from (5d).

8By applying both parametric and nonparametric methods to Japanese data in the 1990s, Ono, Ogawa andYoshida (2004) find that um/uc (= R) has indeed a positive lower bound.

10

u m ( y , m)
= R(π),u c ( y , m)R′(π) > 0.

(18)

It is also assumed that the monetary authority sets R equal to the subjective discount rate whenπ = 0:
R(0) = r(0) = ρ,so that nominal price P will be stabilized when the natural output is realized and the steadystate is reached. In this case the steady state is the same as that under the money stock control
rule, which is given by (9). Obviously, transversality condition (3) is valid there.
The increment of M is transferred to the household sector in a lump-sum manner and thus
μm = τ where μ =M&.
MNote that μ is continuously controlled so that (18) is valid. Given that under (18) real interest
rate r is a function of π:r(π) = R(π) − π,(19)

the monetary policy is called
active when r′(π) > 0,
passive when r′(π) < 0.
From (2), the time derivative of (18) and (19), one derives
Φ y& = − ucmucR′ π& − (ummuc − ucmum)[r(π) − ρ],(20)

where Φ (> 0) is given in assumption 1.

4.1. Original Phillips curve

Substituting into (20) the time derivative of (5a):π& = y& f1′(y/yn)/yn,

and rearranging the result yields⎛ Φ + u cm u c R ′f 1′ / y n ⎞⎟ y& = r(f1(y/yn)) − ρ,− ⎜⎜2 ⎟⎝ [∂ (u m / u c ) / ∂m]u c ⎠(21)

where the coefficient of y& is positive under assumptions 1 and 2 unless ucm is significantly
negative. Thus,>>y& y = 0 ⇔ r′ = 0.
<<

11

Since y is a jump variable, this property shows that under an active monetary policy (viz. r′ > 0)
the dynamics of (21) is unstable and that the equilibrium path is uniquely determined: thesteady state is chosen from the beginning. Under a passive policy (r′ < 0), however, thedynamics is stable and hence there is a continuum of equilibrium paths –i.e., the path isindeterminate.

4.2. Backward-looking adjustment

By substituting (5b) into (20) one obtains
Φ y& = − ucmucR′f2(y/yn) − (ummuc − ucmum)[r(π) − ρ].

(22)Around the steady state given by (9) the characteristic function of the (π, y) dynamics
formulated by (5b) and (22) is
f 2′
yn
= 0.
− u cm u c R ′f 2′
−λ
Φy n

−λ
−

(u mm u c − u cm u m )r ′Φ

It is rewritten as
λ2 + Aλ + B = 0,
where under assumptions 1 and 2<
A = − (λ1 + λ2) = − ucmucR′f2′/(Φy ) = 0>n>⇔ ucm = 0,<><
B = λ1λ2 = r′f2′(ummuc − ucmum)/(Φyn) = 0 ⇔ r′ = 0.
><

From these properties, the two roots satisfyr′ > 0: λ1 > 0 and λ2 < 0;>>>r′ < 0: λ1 = 0 and λ2 = 0 ⇔ ucm = 0<<<

if A2 − 4B > 0,>>
the real part of λ1 and λ2 = 0 ⇔ ucm = 0<<

if A2 − 4B < 0.

Since y is jumpable while π is not, under an active monetary policy (r′ > 0) there is a uniqueequilibrium path. Under a passive policy (r′ < 0) the equilibrium path is non-existent if m and c
are Edgeworth substitutes (ucm > 0) and indeterminate if ucm ≤ 0.

12

4.3. Forward-looking adjustment

Substituting (5c) into (20) yields

Φ y& = − ucmucR′[ρπ − f3(y/yn)] − (ummuc − ucmum)[r(π) − ρ].

(23)

Since r′(π) = R′(π) − 1 from (19), around the steady state given by (9) the characteristic
function of the (π, y) dynamics formulated by (5c) and (23) is
f 3′
yn
= 0.
ucmuc R′f 3′
−λ
Φy n

ρ−λ
−

−

ucmum + ummuc r ′ΦIt reduces to

λ2 + Aλ + B = 0,
where under assumptions 1 and 2
A = − (λ1 + λ2) = − [ρ + ucmucR′f3′/(Φyn)],>B = λ1λ2 = (ucmum − ummuc)r′f3′/(Φyn) = 0<>⇔ r′ = 0.
<

Thus, under an active monetary policy (r′ > 0) the characteristic equation has either two
positive roots or two conjugate complex numbers with a positive real part as long as ucm is not
so significantly negative as to make A positive. Since both y and π are jumpable and the
dynamics is unstable, it has a unique equilibrium path on which the steady state is reached fromthe beginning. Note that this is valid whether the two roots are real numbers or complex
numbers.
Under a passive monetary policy (r′ < 0) it has two real roots among which one is positive
and the other is negative. The equilibrium path is indeterminate since both y and π are jump
variables.9

4.4. Hybrid adjustment

Substituting (5d) into (20) yields

Φ y& = − ucmucR′z − (ummuc − ucmum)[r(π) − ρ],

9

(24)

The result of this subsection is consistent with that of Benhabib et al. (2001a) in the case where real balances
are held for nonproductive purposes.

13where z represents π& , as given by (15). The two equations in (15) and (24) formulate anautonomous dynamic system with respect to π, z and y. Around the steady state given by (9) the
characteristic function is
−λ

1

0
ρ−λ
− (u mm u c − u cm u m )r ′ / Φ − u cm u c R ′ / Φ0− f 4′ / y n = 0.
−λIt is rewritten as

λ3 + Aλ2 + Bλ + D = 0,
where under assumptions 1 and 2
A = − (λ1 + λ2 + λ3) = − ρ < 0,B = λ1λ2 + λ2λ3 + λ3λ1 = −D = − λ1λ2λ3 = −>u cm u c R ′f 4′ <
0
⇔
u
= 0,
=
cm
>
Φy n
<

>
(u mm u c − u cm u m )r ′f 4′ >
=
0
⇔
r
′
= 0.
Φy n
<
<

(25)

In this dynamics m is not jumpable and only two of y, π and z are jumpable under (5d), as
mentioned at the end of section 2.
Under an active policy (r′ > 0) D is positive. Therefore, from the first and third propertiesof (25), one of the three roots is negative and the other two are positive if all the three roots arereal numbers. If one of the three roots is a real number and the other two are complex numbers,from the third property of (25) the real root is negative and hence from the first property of (25)the real part of the complex roots must be positive. Therefore, the equilibrium path is anyway
determinate.
Under a passive monetary policy (r′ < 0) D in (25) is negative. Therefore, when all roots
are real numbers, the case where two of them are positive and the rest is negative is invalid. The
case where all roots are positive and the case where one is positive and the other two are
negative are both possible if ucm < 0, and hence the equilibrium path is either non-existent orindeterminate. If ucm ≥ 0, however, B in (25) is non-positive –i.e., only the latter case is valid
and then the path is indeterminate.
If two of the three roots are mutually conjugate complex numbers, δ ± εi, the third propertyof (25) implies the real root λ to be positive. Since the first and second properties reduce to

λ + 2δ > 0,

14

>
<
2λδ + δ2 + ε2 = 0 ⇔ ucm = 0,><

δ is either positive or negative if ucm < 0 and hence the equilibrium path is either non-existent orindeterminate. If ucm ≥ 0, however, δ is negative and hence the path is indeterminate.
Thus, under a passive monetary policy the equilibrium path is either non-existent orindeterminate if ucm < 0, whether the characteristic roots are all real numbers or not. However,
if m and c are Edgeworth substitutes (ucm > 0) or if the utility function is additive separable in cand m (ucm = 0), the path exists but is indeterminate.
The results of this section are summarized as follows:

Proposition 3. If the monetary authority follows an active interest-rate feedback rule, theequilibrium path is always determinate under any of the four Phillips curves. If a passiveinterest-rate feedback rule is adopted, under any of the four Phillips curves theequilibrium path is never determinate. It is always indeterminate under the original
Phillips curve and the forward-looking one whereas under the backward-looking one andthe hybrid one it is either non-existent or indeterminate.

5. Conclusion
This paper examines equilibrium determinacy under four types of the Phillips curve, viz.
the original, the backward-looking, the forward-looking and the hybrid ones, when themonetary authority adopts a money stock control rule or an interest-rate feedback rule (aTaylor rule). The result is summarized in table 1.
If the monetary authority keeps the nominal money stock to be constant, the equilibrium
path is always determinate under the original Phillips curve and the forward-looking one.
Under the backward-looking Phillips curve and the hybrid one, however, the path can be
non-existent. It is indeed the case if the utility function is additive separable and the elasticityof the marginal utility of money (ηm) is small, as is under a liquidity trap.
The forward-looking Phillips curve is often criticized because of poor empirical fitting.
The original Phillips curve is also criticized because of the lack of a microeconomic foundationand empirical fitting. However, as is seen from figure 2, the original one seems to fit the15

Japanese data quite well.10 Such an empirical analysis is beyond the scope of the present paper
but the present theoretical result and figure 2 may suggest that we should reexamine theempirical validity of the original Phillips curve and explore a microeconomic foundation for
it.11
Turning to an interest-rate feedback rule, if the monetary policy is passive, the equilibrium
path is never determinate under any of the four Phillips curves: it is either non-existent orindeterminate. If it is active, the equilibrium path is always determinate under any one of them.
Thus, the monetary authority must actively control the interest rate in order for the equilibrium
path to be determinate, as long as it follows a Taylor rule.

Appendix: Proof of Proposition 2
The case of the backward-looking Phillips curve:

From (9), (10), (12) and (13) in which ucm = 0, the characteristic function is thenF(λ) = λ3 − ρλ2 + (f2′/ηc)λ − ρηmf2′/ηc = 0,

(A1)

where ηc = − uccc/uc and ηm = − ummm/um. As proven in subsection 3.2, if it has two complex
roots (δ ± εi), the real root λ1 is positive. Since the first equation of (13) and (A1) yieldρ = λ1 + 2δ,the common real part of the complex roots δ satisfies>>
δ = 0 ⇔ ρ = λ1 (> 0).
<
<

(A2)

Since λ1 is the only real root in the present case, function F(⋅) given by (A1) intersects thehorizontal axis only once and from below. Moreover, since F(ρ) is><
F(ρ) = (ρf2′/ηc)(1 − ηm) = 0 ⇔ ηm = 1,><

and F(λ1) = 0 by the definition of λ1, using (A2) one finds that δ satisfies

10

Using Japanese quarterly data (1981-2000) Kitaura et al. (2002) find that the backward-looking Phillips
curve is empirically rejected and that the original Phillips curve fits the data well. The yearly data used in figure 2
are consistent with it, as shown by figure 3 –i.e., there is almost no correlation between a yearly change in the
inflation rate and the unemployment rate.
11

Akerlof et al. (1996) is an important attempt to present such a foundation.

16

>
<
δ = 0 ⇔ ηm = 1.
>
<

(A3)

Next, it is shown that (A1) indeed has only one real root if ηm > 1. From (A1),F′(λ) = 3λ2 − 2ρλ + (f2′/ηc).

Therefore, if ρ2 − 3f2′/ηc > 0, the two extrema of F(λ) exist and the local maximum isF((ρ −

ρ 2 − 3 f 2′ / η c )/3)

= − [2(ρ2 − 3f2′/ηc)(ρ − ρ 2 − 3 f 2′ / η c ) + 24ρf2′/ηc]/33 − (ηm − 1)ρf2′/ηc,which yields
F((ρ −

ρ 2 − 3 f 2′ / η c )/3) < 0 if ηm > 1.

This property shows that there is indeed only one real root if ηm > 1 and ρ2 − 3f2′/ηc > 0. If ρ2 −3f2′/ηc < 0, F(λ) has no extremum and hence there is only one real root, whether ηm > 1 or not.
Therefore, if ηm > 1, (A1) has only one real root, which is λ1 (> 0). From (A3) δ is negative inthis case and hence the path is determinate.
The above result also implies that ηm must be less than 1 in order for the three roots of (A1)
to be real numbers and then the path is non-existent, as shown in subsection 3.2. Furthermore,
if two of the three roots are complex numbers and λ1 (> 0) is the only real root in the case whereηm < 1, from (A3) δ is positive –i.e., the path is non-existent. Thus, if ηm < 1, the path isnon-existent, whether (A1) has two complex roots or not.
If ηm = 1, δ is zero as shown by (A3) and hence the characteristic roots are a positive real
number (viz. λ1) and two pure imaginary ones. Since y is the only jump variable, it jumps toone of the cyclical paths. Although the steady state given by (9) is not reached along any
cyclical path, all variables cyclically move within some finite ranges and transversality
condition (3) is valid. Therefore, the path is determinate.

The case of the hybrid Phillips curve:

Since it has already been found in subsection 3.4 that the path is non-existent when all
roots are real numbers and determinate when all roots are complex numbers, the case where
two roots are real numbers, λ1 and λ2, and the other two are complex numbers, δ ± εi, is
examined. Note that the path is non-existent if and only if δ > 0 since λ1 and λ2 satisfy

λ1 > 0, λ2 > 0,17

as shown in subsection 3.4.
From (9), (10), (12) and (16) in which ucm = 0, the characteristic function isG(λ) = λ4 − 2ρλ3 + ρ2λ2 − (f4′/ηc)λ + ρηmf4′/ηc = 0.

(A4)

Since λ1 and λ2 are real roots of (A4), from the first property of (17) and (A4) they satisfy

λ1 + λ2 = 2(ρ − δ) > 0,
(λ1 − ρ)[λ12(λ1 − ρ) − f4′/ηc] = (f4′/ηc)(1 − ηm)ρ,(λ2 − ρ)[λ22(λ2 − ρ) − f4′/ηc] = (f4′/ηc)(1 − ηm)ρ.

(A5)

By taking the difference between the second and third equations above and substituting thefirst equation into the result one obtains
f4′/ηc + 2δ (ρ − 2δ)2 = 2(ρ − 2δ)[(ρ − 2δ)ρ − λ1λ2].

(A6)Since from the first equation of (A5) λ1λ2 satisfies

λ1λ2 = (λ1 + λ2 − ρ)ρ − (λ1 − ρ)(ρ − λ2) = (ρ − 2δ)ρ − (λ1 − ρ)(ρ − λ2),
substituting this value into λ1λ2 in (A6) and rearranging the result gives(λ1 − ρ)(ρ − λ2) = [f4′/ηc + 2δ (ρ − 2δ)2]/[2(ρ − 2δ)].

(A7)

Multiplying the second equation of (A5) by (λ2 − ρ) and the third one by (λ1 − ρ),summing them up and substituting the first equation into the result yields(λ1 − ρ)(ρ − λ2)[(ρ − 2δ)2 + (λ1 − ρ)(ρ − λ2)] = (1 − ηm)ρf4′/ηc.
Substituting (λ1 − ρ)(ρ − λ2) given by (A7) into the above equation and rearranging the result
leads to2ρ3(1 − 2ηm) − f4′/ηc = 4δ(ρ − δ)[2ρ(1 − 2ηm) + (ρ − 2δ)4/(f4′/ηc)].

(A8)

If 2ρ3(1 − 2ηm) − f4′/ηc > 0, then 1 − 2ηm > 0. From the first property of (A5), ρ − δ > 0. Using
these properties and (A8) one finds δ > 0 if 2ρ3(1 − 2ηm) − f4′/ηc > 0.
As the converse, if δ > 0, then 1 − 2ηm > 0 since (A8) is rewritten as2ρ(ρ2− 2δ)2(1 − 2ηm) = 4δ(ρ − δ)(ρ − 2δ)4/(f4′/ηc) + f4′/ηc.
Therefore, the right-hand side of (A8) is positive so that the left-hand side is also positive, i.e.,2ρ3(1 − 2ηm) − f4′/ηc > 0.
In sum,δ > 0 ⇔ 2ρ3(1 − 2ηm) − f4′/ηc > 0,(A9)and then the path is non-existent, as long as (A4) has two real and two complex roots.
Next, it is shown that the case of four complex roots does not arise in the area given by(A9). G(λ) given by (A4) satisfies
G(ρηm) = [ηm(ηm − 1)]2ρ4 > 0,
18

G(ρ) = (f4′/ηc)(ηm − 1)ρ < 0 if ηm < 1.

(A10)

When the condition given by (A9) is valid, ηm < 1/2 and hence the two properties of (A10)show that G(λ) has at least one real root. Thus, the case of four complex roots (in which thepath is determinate) does not arise in the area given by (A9). Either the case with two real andtwo complex roots or the case of four real roots occurs.
The area in which the four roots are real numbers is included by the area given by (A9), as
shown below. From (A4),G′(λ) = 2λ(2λ − ρ)(λ − ρ) − f4′/ηc,G″(λ) = 2[λ − (1/2 −3 /6)ρ][λ − (1/2 +3 /6)ρ].

(A11)

Since ηm < 1/2 in the area given by (A9), from (A10) and (A11) G(ρ) satisfies
G(ρ) < 0, G″(λ) > 0 for ∀λ > ρ,

implying that there is a positive real root that is larger than ρ and that there is no other real root
larger than it. Therefore, from (A11), if the four roots are real numbers,
G(θ1) < 0, G(θ2) > 0,

where G′(θ1) = G′(θ2) = 0, 0 < θ1 < (1/2 −3 /6)ρ < θ2 < ρ.

(A12)

Note that 0 < θ1 since all the four roots are positive, as shown in subsection 3.4. Since thesecond property of (A11) implies
G″(λ) > 0 for ∀λ < (1/2 −3 /6)ρ,using the first equation in (A11) and (A12) one finds
0 = G′(θ1) < G′((1/2 −3 /6)ρ) = ( 3 /9)ρ3 − f4′/ηc,i.e., f4′/ηc must satisfy
f4′/ηc < ( 3 /9)ρ3.

(A13)

Furthermore, from the first equation in (A11) and (A12),
G(θ1) < 0, G′(θ1) = 2θ1(2θ1 − ρ)(θ1 − ρ) − f4′/ηc = 0.

Therefore, replacing f4′/ηc by 2θ1(2θ1 − ρ)(θ1 − ρ) in (A4) and rearranging the result yields
G(θ1) = θ1(ρ − θ1)H(θ1) < 0,where H(θ1) = 3θ12 − (1 + 4ηm)ρθ1 + 2ηmθ12.
Since θ1 < ρ from (A12), the above property shows H(θ1) to be negative and hence θ1 is locatedbetween the two solutions that makes H(θ1) = 0:
[1 + 4ηm − (16ηm2 − 16ηm + 1)1/2]ρ/6 < θ1 < [1 + 4ηm + (16ηm2 − 16ηm + 1)1/2]ρ/6, (A14)where ηm satisfy
19

16ηm2 − 16ηm + 1 ≡ 16[ηm − (2 −Since θ1 < (1/2 −3 )/4][ηm − (2 +3 )/4] > 0.

3 /6)ρ, as shown in (A12), in order for θ1 that satisfies (A14) to exist ηmmust satisfy[1 + 4ηm − (16ηm2 − 16ηm + 1)1/2]ρ/6 < (1/2 −3 /6)ρ.

Since this property reduces to{[ηm − (2 −3 )/4][ηm − (2 +3 )/4]}1/2 > ηm − (2 −3 )/4,

ηm must satisfy
ηm < (2 −3 )/4.

(A15)

Therefore, if the four roots are real numbers, both (A13) and (A15) are valid.
When f4′/ηc and ηm satisfy (A13) and (A15) respectively, it is easily found that thecondition given by the right-hand side of (A9) is valid. Thus, the area of four real roots is
included by the area given by (A9). Since the path is non-existent when all roots are real
numbers, as shown in subsection 3.4, and (A9) gives the condition for the path to be
non-existent in the case of two complex roots, (A9) in fact shows the condition under which thepath is non-existent, whether all roots are real numbers or not.

20

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23

Original

Backward-looking

Forward-lookin

Hybridg
Fixed money Determinate

Non-existent

Determinate

Non-existentstock

or determinate

Active Taylor Determinate

Determinate

Determinate

Determinate

Indeterminate Non-existentIndeterminate

Non-existentor determinate

rule
Passive
Taylor ruleor indeterminate

or indeterminate

Table 1: Equilibrium Determinacy under Various Phillips Curves

24

ηm

1/2

O

2ρ

3

f4′/ηcFigure 1: ηm and f4′/ηc That Make the Path Non-existent25

CPI inflation rate (%)
4
91
3

90
89

97

2
92
93

1

94

88
0

98
96

87
0

1

2

3 95

06
4 05

04
9 95
00 01

-1

03

6

02

-2

Unemployment rate (%)
Statistics Bureau, Ministry of Internal Affairs and Communications, Japan.
Unemployment rate: http://www.stat.go.jp/data/roudou/longtime/03roudou.htm.
CPI: http://www.stat.go.jp/data/cpi/200707/index.htm
and http://www.stat.go.jp/data/cpi/longtime/index.htm.

Figure 2: Phillips Curve in Japan (1987-2006)26

A yearly change in CPI inflation rate
2
97

89
1

90

0

1

2

93

8 73
94

-1

03

04

96

91

0

06

88

01
4

05

95

5
00

02 6

99
98

-2

92

Unemployment rate (%)
Statistics Bureau, Ministry of Internal Affairs and Communications, Japan.
Unemployment rate: http://www.stat.go.jp/data/roudou/longtime/03roudou.htm.
CPI: http://www.stat.go.jp/data/cpi/200707/index.htm
and http://www.stat.go.jp/data/cpi/longtime/index.htm.

Figure 3: Change in Inflation and Unemployment Rate

27