Discussion Papers In Economics
And Business

Equilibrium Dynamics in Discrete-Time Endogenous
Growth Models with Social Constant Returns
Kazuo Mino
Kazuo Nishimura
Koji Shimomura
Ping Wang

Discussion Paper 05-34

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANEquilibrium Dynamics in Discrete-Time Endogenous
Growth Models with Social Constant Returns
Kazuo Mino
Kazuo Nishimura
Koji Shimomura
Ping Wang

Discussion Paper 05-34
December 2005
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Equilibrium Dynamics in Discrete-Time Endogenous
Growth Models with Social Constant Returns
Kazuo Mino, Osaka University
Kazuo Nishimura, Kyoto University
Koji Shimomura, Kobe University
Ping Wang, Washington University in St. Louis and NBER
December 2005
Abstract: The existing literature establishes possibilities of local determinacy anddynamic indeterminacy in continuous-time two-sector models of endogenous growthwith social constant returns. The necessary and suﬃcient condition for local determinacy is that the factor intensity rankings of the two sectors are consistent in theprivate/physical and social/value sense. The necessary and suﬃcient condition for dynamic indeterminacy is that the final (consumable) good sector is human (pure) capitalintensive in the private sense but physical (consumable) capital intensive in the social
sense. This paper re-examines the dynamic properties in a discrete-time endogenous
growth framework and finds that conventional propositions obtained in continuous timeneed not be valid. It is shown that the established necessary and suﬃcient conditions on
factor intensity rankings for local determinacy and dynamic indeterminacy are neither
suﬃcient nor necessary, as the magnitudes of time preference and capital depreciation
rates both play essential roles. (JEL Classification: E32, J24, O40)
Keywords: Sectoral Externalities, Endogenous Growth, Dynamic Determinacy/Indeterminacy.
Acknowledgment: We have benefitted from discussion with Robert Becker, Eric Bond,
Michael Kaganovich, Karl Shell and participants of the Midwest Macroeconomic Conference in Chicago and the Midwest Economic Theory and International Trade Meetings at
Indiana University. The fourth author acknowledges financial support from the Institute of
Economics and Business Administration of Kobe University and the Institute of Economic
Research of Kyoto University to enable this international collaboration. The usual disclaimer
applies.
Correspondence: Kazuo Mino, Faculty of Economics, Osaka University, 1-7 Machikaneyama,
Toyonaka, 560-0043 Japan, E-mail: mino@econ.osaka-u.ac.jp

1

Introduction

In the past decade, there has been a growing literature studying under what circumstances dynamic indeterminacy may occur. When dynamic indeterminacy arises, theperfect-foresight equilibrium is at least locally indeterminate in the sense that there
are multiple (often a continuum of) converging transition paths. The case of indeterminacy has created vast interest due to the observations that many macro variables
are volatile over time and that economic growth rates are dispersed across countries.
It is worth emphasizing that most of the applied papers focusing on calibrated comparative dynamics or policy simulations use discrete-time models in which there exists
a unique trajectory satisfying saddle-path stability (local determinacy). In contrast,
the theoretical researches on dynamic indeterminacy has been mostly based on thecontinuous-time setting due to its tractability. This is particularly true for the studyof equilibrium indeterminacy in endogenous growth models. In fact, to our knowledge, no one has undertaken a complete examination of the validity of continuous-time
dynamics properties under discrete time in an endogenously growing economy whereindeterminacy may be present.1 This paper provides the first attempt at such an
endeavor.
Within the endogenous growth framework á la Romer (1986) and Lucas (1988),
previous studies conclude that dynamic indeterminacy may occur when the model
economy involves external increasing returns or distortionary taxes. The early literature, best represented by Benhabib and Perli (1994), Boldrin and Rustichini (1994),
and Xie (1994), focuses on increasing returns as a result of uncompensated positive
spillovers.2 More recent work, instead, considers overall constant-returns production
technologies, which the present paper follows.3 The analysis of the multi-sector dynamical system with perpetual growth has been viewed rather complex, because the1

Dynamic properties have been examined in discrete-time exogenous growth models with sector-

specific externalities. The reader is referred to Benhabib, Nishimura and Venditti (2002) and papers
cited therein. In a companion paper, Nishimura and Venditti (2002) revisit the issue with intersectoral
externalities.
2
See a survey by Benhabib and Rustichini (1994). It should be noted that dynamic indeterminacy
may also arise in other types of growth models with increasing returns or matching externalities, suchas Benhabib and Farmer (1994), Farmer and Guo (1994) and Laing, Palivos and Wang (1995).
3
For considerations of sector-specific externalities with social constant returns, the reader is referred
to Benhabib and Nishimura (1998, 1999), Benhabib, Meng and Nishimura (2000) and Mino (2001).

conventional algorithm transforms the nonstationary system into stationary ones byconstructing ratios of growing quantities and then studying their dynamic properties
in conjunction with factor allocation variables.4 Yet, as pointed out by Bond, Wang
and Yip (1996), the analysis is greatly simplified if the transformation takes place byseparately characterizing the price and the quantity dynamics using both the primal
and the dual. The transformed prices can be in forms of either relative prices (asin Bond, Wang and Yip, 1996, and Mino, 2001) or nominal prices (as in Benhabib,
Meng and Nishimura, 2000). Under this transformation, price dynamics is determinedby prices alone and thus the dynamical system is recursive. Moreover, the dynamic
properties are found to depend exclusively on the factor intensity rankings of sectoral
productions.
More specifically, under private constant returns without externalities but with distortionary factor taxes, Bond, Wang and Yip (1996) establish the polarization theorem:
when factor tax distortion is moderate such that the factor intensity rankings in thephysical and value sense are consistent, the price and quantity dynamics are polarized— when one is stable, another must be unstable, yielding saddle-path stability.5 They
also show that dynamic indeterminacy can arise if the final good sector is human capital intensive in the physical sense but physical capital intensive in the value sense.
While the former factor intensity ranking ensures stable quantity dynamics, the latter
guarantees stable price adjustments, thus resulting in locally indeterminate transition.
In another class of endogenous growth models, production exhibits social constant returns with positive externalities in the absence of distortionary factor taxation. As
shown by Benhabib, Meng and Nishimura (2000) and Mino (2001), the polarization
theorem for local determinacy holds, whereas dynamic indeterminacy emerges when
the final good sector is human capital intensive in the private sense but physical capitalintensive in the social sense. Notably, a general property can be established in thatthe factor intensity ranking in the private sense corresponds to that in the physicalsense whereas the factor intensity ranking in the social sense corresponds to that in thevalue sense. Thus, one can easily reconcile the findings regarding local determinacy
and dynamic indeterminacy in Bond, Wang and Yip (1996) and those in Benhabib,
Meng and Nishimura (2000) and Mino (2001).64
5

See Benhabib and Perli (1994), among many others.
While Bond, Wang and Yip (1996) examine both capital and labor income taxes, Mino (1996)

studies the eﬀects of capital taxation.
6
Similar propositions can be extended to exogenous growth models (e.g., Benhabib and Nishimura,2

This paper considers a class of endogenous growth models with social constantreturns in the presence of positive externalities. The departure from the literaturesummarized above is that time is discrete. Our main contribution is to show that conventional propositions obtained in continuous-time models are no longer valid. First,
the balanced-growth path need not be locally determinate even if the factor intensity
rankings in the private and social sense are consistent. Second and most interestingly,
even under the conventional necessary and suﬃcient conditions that the final good
sector is human capital intensive in the private sense and physical capital intensivein the social sense, dynamic indeterminacy need not emerge unless the rates of time
preference and capital depreciation are suﬃciently small and the rates of depreciationsof the two capitals are suﬃciently close. Finally, even if the conventional necessary andsuﬃcient conditions fail to hold, dynamic indeterminacy may still arise when the final
good sector is human capital intensive in both private and social sense. These findings are powerful because if one desires to conduct calibrated comparative dynamics
or policy simulations under local determinacy, the parameters must be so chosen thatdynamic indeterminacy or dynamic instability would not occur.7
The remainder of the paper is summarized as follows. In Section 2, we describe
the basic environment of the model economy by specifying the economic agents, theirpreferences and capital endowments, the production technologies and the optimizing
behavior. Section 3 defines the concepts of dynamic competitive equilibrium and balanced growth path and establishes the existence and uniqueness of the balanced growthpath. In Section 4, we characterize the local dynamics of the system. Finally, Section
5 concludes the paper and provides possible avenues for future work.
1998, 1999) and open-economy endogenous growth models (e.g., Nishimura and Shimomura, 2002, andBond, Trask and Wang 2003).
7
It has been known that time frequency may play an important role for the stability of dynamicalsystems. For example, in the optimal exogenous growth literature that treats discrete-time dynamic
systems, Mitra (1998) and Baierl, Nishimura and Yano (1998) show that levels of capital depreciation
rates as well as time discount rate aﬀect the stability conditions, while the dynamic properties are
generally independent of these parameters in continuous-time models. Similarly, in her study ofcalibrated, discrete-time models of real business cycles with market distortions, Schmitte-Grohé (1997)
reveals that the conditions for indeterminacy of equilibrium involve the magnitude of capital depreciate
rate, which plays no role in the continuous-time counterparts. As summarized above, the present
study also confirms the relevance of time frequency in the context of endogenous growth models withexternalities and social constant returns.

3

2

The Basic Model

2.1

Production Technology and Capital Formation

We consider a discrete-time model (indexed by t = 0, 1, 2, ...) of endogenous growthwith two perpetually accumulated reproducible capitals (denoted Xi , i = 1, 2) and two
distinctive sectors (labeled by j = 1, 2). Let the first reproducible capital be consumable and the second be non-consumable. For illustrative purposes, we may refer X1
as physical capital and X2 as human capital. Thus, sector 1 produces the final goodfor consumption and physical capital investment, whereas sector 2 produces “education services” that enhance human capital. The final good and education services areproduced by both physical and human capitals.
Following the setup of the continuous-time framework by Benhabib, Meng andNishimura (2000), we consider that both capitals generate sector-specific, positive external eﬀects and that both production technologies exhibit social constant returns.
More specifically, denoting Yjt as the output of sector j in time t and vijt as the fraction of factor Xit (measured at the beginning of period t) devoted to sector j productionP( 2j=1 vij = 1), the production technologies with time-to-build is given by,
Yjt = αj

2
Y
i=1

¡
¢bij
(vijt Xit )β ij vijt Xit,(1)

where sector-specific externalities are captured by the society’s average values of sectorspecific factors (labeled with bars), αj > 0 is a scaling factor, 0 < β ij < 1 andβ 1j +β 2j < 1. Thus, b1j and b2j , respectively, measure the degree of positive externalities
from physical and human capital in sector j. The private returns to scale of theP
two reproducible factors are measured by 2i=1 β ij , whereas the social returns are by
P2 b
P2 b
P2
i=1 (β ij + bij ) ≡i=1 β ij . By assumption, constant social returns imply
i=1 β ij = 1
for j = 1, 2. As to be formally defined later, we have vijt Xit = vijt Xit in equilibrium.
Let Ct denote period t consumption of the final good and δ i > 0 denote the (constant) depreciation rate of capital Xi ). The evolution of the two capital stocks cantherefore be expressed as (t = 0, 1, 2, ...):
X1,t+1 = Y1t + (1 − δ 1 )X1t − Ct ,(2)

X2,t+1 = Y2t + (1 − δ 2 )X2t ,(3)

where the initial stocks of both capitals, X10 and X20 , are historically given and positive.
4

2.2

Producer Optimization and Factor Market Equilibrium

The infinite-horizon problem of firms discounted by the market interest rate under
our competitive equilibrium framework can be reduced to a simple period optimization problem. Specifically, each competitive firm in sector j chooses factor allocation{vijt Xit } and output to maximize its period profit given output price Pj and input
prices {W1 , W2 }:π jt = Pjt Yjt −2
X

Wit vijt Xit

i=1

subject to the production technologies specified as in (1). The first-order conditionsare (i, j = 1, 2 and t = 0, 1, 2, ...):
Pjt β ijt

Yjt
= Wit .
vijt Xit(4)v

X

The cost minimizing solution of vijt Xit is given by aij Yjt , where aij ≡ ijtYjt it can bereferred to as the input coeﬃcients (which is time-invariant under the time-invariant
production technologies specified above). From (4), we have:
aij = β ij Pjt Wit−1 .

(5)This can be combined with the production technologies (1) under the equilibrium
condition, vijt Xit = vijt Xit , to yield:
Pjt = α−1
j

2
Y
¡ −1
¢βb
β hj Wht hj .

(6)

h=1

Utilizing the property of constant social returns

2P
bij = 1,we can substitute (6) intoβi=1(5) to eliminate output prices:aij = α−1
j

2
Y

h=1

Ã −1
!βb hjβ hj Wht
β −1
ij Wit.

(7)

It is convenient to express in matrix form. Let A ≡ (aij ) , Yt ≡ (Y1t , Y2t )0 and
Xt ≡ (X1t , X2t )0 , where the superscript “0 ” denotes the transpose operation. Then, full2
P
employment of both factor inputs requires
aij Yjt = Xit , or, equivalently,
j=1

AYt = Xt .
5

(8)bij /β ij , Ab ≡ (aij ), P ≡ (P1 , P2 )0 and W ≡ (W1 , W2 )0 . Let Bb
Next, define baij ≡ aij βbij (i, j = 1, 2) and Iw and Ip be diagonal matrices withbe a matrix with elements βdiagonal elements (w1 , w2 ) and (p1 , p2 ), respectively. It can be easily seen from (5) thatbij = baij Wit , or, in matrix form,
Pjt β
Since

b = (A)I
b w.
Ip B

(9)

2
2
P
bij = 1, we obtain: Pj = P Wib
aij , or, in matrix form,
β

i=1

i=1

b 0 Wt ,Pt = (A)(10)

which can be conveniently referred to as the competitive profit condition. ¿From (6),
we have a pair of Samuelsonian relationships that relate factor prices to output prices
under factor price equalization:
³
´
b
b
b
b
log P1 α1 = β 11 log W1 + β 21 log W2 − β 11 log β 11 + β 21 log β 21³´b12 log W1 + β
b22 log W2 − βb12 log β 12 + β
b22 log β 22 .
log P2 α2 = βb ≡Define ∆ ≡ β 11 β 22 − β 12 β 21 as the private factor share determinant and ∆b22 − βb21 as the social factor share determinant. While the sign of ∆ indicates
b12 β
b11 β
β
b gives such a
the factor intensity ranking in the private (physical) sense, the sign of ∆ranking in the social (value) sense. Consider that the two sectors have diﬀerent factor
intensity rankings:
b 6= 0.
Assumption 1. ∆ 6= 0 and ∆Under this assumption, we can invert the Samuelsonian relationships to solve Wi (i =1, 2) as a function of {P1 , P2 }:bβW1t = w1 P1t22

b
/∆

b
−β

P2t 21b/∆b−β; W2t = w2 P1t 12

b
/∆

b
β

P2 11

b
/∆

,

(11)

where
´βb 22 /∆b ³´−βb 21 /∆b
³
b
b
b
b
β
β
β
β
α2 β 1212 β 2222
α1 β 1111 β 2121´−βb 12 /∆b ³´βb 11 /∆b
³
b
b
b
b
β
β
11 β 2112 β 22α2 β 12 β 22
= α1 β 11 β 21
.

w1 =
w2

These expressions yield the standard Stolper-Samuelson property: how factor prices
are related to output prices depends crucially on the factor intensity ranking in thebvalue sense, i.e., the sign of the social factor share determinant ∆.
6

Remark 1. Our assumption of decreasing returns in the private technology withrespect to capital inputs can be consistent with the free entry and zero-excess profit
conditions. Suppose that, in addition to X1j and X2j , each production sector employes
a sector-specific input denoted by X3j (j = 1, 2). We assume that the supply of
X3j is fixed and that it has negative external eﬀects because of congestion. Lettingβbαj = X3j3j X̄3j3j in (1), we assume that Σ3i=1 β ij = 1, β ij > 0 (j = 1, 2) and Σ3i=1 bij = 0.
Notice that since X1 and X2 are assumed to have positive externalities, Σ3i=1 bij = 0
means that b3j has a negative value. Given these assumptions, both social and private
technologies exhibit constant returns to scale with respect to X1j , X2j and X3j , sothat firms cannot earn excess profits under free entry. Now, suppose further thatb3 = −β 3 (< 0) . Then (1) satisfies constant and decreasing returns to scale in X1j and
X2j from the social and private perspectives, respectively.

2.3

Household Optimization

The representative household has a time-additive preference with a subjective discountfactor ρ ∈ (0, 1) and a constant elasticity of intertemporal substitution σ −1 > 0. Her
lifetime utility is thus given by:
U=

∞
X
t=0

ρt

Ct1−σ − 1
. σ > 0,1−σσ 6= 1.

Let Πt be the distribution of profits from both sectors to the representative agent. Denote Wit and Pjt (i, j = 1, 2) as the nominal factor prices and nominal good/education
service prices of the ith capital and jth output at time t where all prices are measured in utils without accounting for intertemporal variation from time-discounting.
Assuming that both capitals are owned by households (of mass one), the flow budget
constraint facing the presentative household at time t can be written as:
W1,t X1,t + W2,t X2,t + π 1t + π 2t
= P1t Ct + P1t [X1,t+1 − (1 − δ 1 )X1,t ] + P2t [X2,t+1 − (1 − δ 2 )X2,t ] ,(12)where the lefthand side is the sum of factor incomes and profit redistributions and therighthand side is the total expenditure (consumption plus gross investments in physicaland human capitals).88

If we follow the implication pointed out in Remark 1 in Section 2.1, we may denote π j = W3 X3ji

(j = 1, 2), where W3 is the price of X3j and it satisfies W3 = Pj β 3j Yj /X3j .

7

Therefore, the representative household chooses {Ct , X1,t , X2,t } to maximize thelifetime utility U subject to the budget constraint (12). Since input and output prices
are defined in utils without time-discounting, the Lagrangean function can be specified
as:
L =

∞
X
t=0

ρt {

Ct1−σ − 1
+ W1t X1t + W2t X2t + π 1t + π 2t
1−σ−P1t Ct − P1t [X1,t+1 − (1 − δ 1 )X1t ] − P2t [X2,t+1 − (1 − δ 2 )X2t ]}.
The first-order necessary conditions are (i = 1, 2):¡¢∂L= ρt Ct−σ − P1t = 0,∂Ct
∂L= ρt [Wit + (1 − δ 1 )Pit ] − ρt−1 Pi,t−1 = 0,
∂Xit

(13)
(14)

which are suﬃcient for optimization under the transversality conditions given by (i =1, 2):lim ρt Pit Xi,t+1 = 0.
t→∞3

Equilibrium

Within our endogenous-growth framework with sector-specific externalities, a dynamic
competitive equilibrium satisfies the following conditions:(i) Each consumer maximizes her life-time utility subject to the flow budget constraint (4) under a given sequence of perfectly anticipated prices and profits,{Wi,t , Pi,t , π i,t }∞t=0 (i =1,2). Additionally, the consumer’s optimal plan stisfies thetransversality condition: limt→∞ ρt Pit Xi,t+1 = 0.
(ii) Each firm maximizes its profits in each moment by taking external eﬀects, vij,ti Xi,t(i, j = 1, 2), as given.
(iii) The goods market clearing conditions, (2) and (3), always hold.
(iv) Both capitals are fully employed: AYt = Xt and Σ2j−1 vij = 1 (i, j = 1, 2) for allt ≥ 0.
(v) The external eﬀects satisfy the consistency conditions so that vij,ti Xi,t = vij,ti Xi,t (i, j =1, 2) for all t ≥ 0.
8

Notice that under the budget constraint, the production technology, the capital evolution equations, the factor demand conditions, and the full employment conditions, one
if the goods market clearing conditions is redundant, i.e., the Walras’ law holds.
Since both capitals are accumulated perpetually, the quantity and price variables
stated above are nonstationary. A balanced growth path (BGP) is a dynamic competitive equilibrium in which quantities and nominal prices {Xit , Yjt , Ct , Wit , Pjt } grow atsome constant rates and factor allocation shares are time-invariant. A BGP is called
nondegenerate if the rates of growth of quantities {Xit , Yjt , Ct } are positive.

3.1

Transformation of the Dynamical System

To characterize the dynamical system, it is important to properly transform the nonstationary system into a stationary one. Given the property of constant social returnsof the production technologies and the time additive preference with constant elasticity of intertemporal substitution, it is straightforward from (1)-(3), (13) and (6) that,along a BGP, factor inputs (Xi ), outputs (Yj ) and consumption (C) will all grow at acommon rate, denoted g, whereas factor prices (Wi ) and good/education service prices(Pj ) will decrease at a common rate, (1 + g)σ . To transform these endogenous variables
into stationary variables, we therefore discount quantities by dividing by (1 + g)t andupcount prices by multiplying (1 + g)σt :xt ≡ Xt (1 + g)−t ; yt ≡ Yt (1 + g)−t ; ct ≡ Ct (1 + g)−t ;pt ≡ Pt (1 + g)σt ; wt ≡ Wt (1 + g)σt ,where all variables in lower case denote the corresponding stationary values after transformation. The transversality conditions require:ρ (1 + g)1−σ < 1.

(15)

(1 + g) xt+1 = yt + (I − Iδ ) xt − zt,(16)

Thus, (2), (3) and (13) imply:

while updating (14) by one period yields:(1 + g)σ pt = ρ (I − Iδ ) pt+1 + ρwt+1 ,

9

(17)

³
´0
−1/σ
where zt ≡ (ct , 0)0 = p1t , 0 and Iδ is a diagonal matrix with diagonal elements,
δ 1 and δ 2 . Moreover, we can rewrite (8) and (10) in terms of transformed stationary
variables:(18)
Ayt = xt
wt = (Â0 )−1 pt .

(19)

Finally, substituting (18) and (19) into (16) and (17), we obtain the two fundamental equations governing the dynamical system in {x1t , x2t , p1t , p2t }:(1 + g) xt+1 = [(I − Iδ ) + A−1 ]xt − zt

(20)

ρ[(I − Iδ ) + (Â0 )−1 ]pt+1 = (1 + g)σ pt .

(21)

It is important to note that once the balanced growth rate, g, is solved, these two
fundamental equations alone determine the dynamics of normalized capital inputs andnormalized nominal output prices (x, p). Since there is no money illusion, this system
in nominal prices must exhibit one-dimensional linear dependency, containing a unit
root (to be shown later).

3.2

Balanced Growth Path

Along a BGP, the transformed variables are stationary and hence, (20) and (21) imply:£ −1¤
A − (gI + Iδ ) x = z
∙
´0 ¸
³
σ
−1
b
(1 + g) I − ρ I − Iδ + A
p = 0.

Since prices are positive, (23) alone determines g by,
∙
´0 ¸
³
σ
−1
b
det (1 + g) I − ρ I − Iδ + A
= 0.

(22)
(23)

(24)

Upon pinning down the balanced growth value of g, x can be determined by inverting(22), provided that det [I − A (gI + Iδ )] 6= 0 (to be shown later):x = [I − A (gI + Iδ )]−1 Az.

(25)

For convenience, define si (g) ≡ (1 + g)σ − ρ (1 − δ i ). We can then write (23) as:
b−1 p,Is p = ρA
10(26)

where Is is a diagonal matrix with diagonal elements si (i = 1, 2).
The remainder of this subsection is devoted to proving the existence and uniquenessof the BGP by examining normalized prices and quantities as well as the balanced
growth rate, to which we now turn.
3.2.1

Existence and Uniqueness of Normalized Prices and Balanced Growth
Rate

As in the continuous-time model, the balanced growth rate and price variables are
pinned down in the system in a recursive manner prior to the determination of thequantity variables. For the BGP to be nondegenerate and to satisfy the transversality
condition, it must hold that max {1, ρ (1 + g)} < (1 + g)σ . If σ ≥ 1, this is satisfied
for any g > 0 (because ρ < 1). If 0 < σ < 1, then it is equivalent to the following
conditions:(27)
1 < (1 + g)σ < ρ−σ/(1−σ) .
This resembles the Jones-Manuelli conditions (first inequality) and the Brock-Galecondition (second inequality). Upon solving for the balanced growth rate g ∗ , thiscondition can be rewritten in terms of primitives. We impose the following conditions
on α1 and α2 :s1 (ρ−1/(1−σ) −1)
s2 (ρ−1/(1−σ) −1)
s1 (0)
s2 (0)
<
α
<
and
<
α
<
.
Assumption 2.
1
2
b
b
b
b
b
b
b
b
β
β
β
β
β
β
β
11
21
11
21
12
22
12 β 22
ρβ 11 β 21

ρβ 11 β 21

ρβ 12 β 22

ρβ 12 β 22b −1 Ip−1 , which can beb−1 = Iw B
Denote 1 ≡ (1, 1)0 . Recall from (9) that we have: A
substituted into (26) to yield,b −1 Ip−1 p = ρIw B
b −1 1 = ρIw ,Is p = ρIw B(28)

si (g)pi = ρwi .

(29)

b = 1 and hence,
where in deriving the last equality we have used the property, B1
b −1 1 = 1. This implies, for i = 1, 2,
B
From (11), we can express real factor prices wi /pi as functions of relative prices q ≡p1 /p2 and plug the results into (29) to obtain:bb

ρw1 qβ 21 /∆ − s1 (g) = 0,bbρw2 q −β 12 /∆ − s2 (g) = 0(30)

We begin by showing that under Assumption 1, Assumption 2 is suﬃcient for thebalanced growth rate g∗ to satisfy (27):11Lemma 1. Under Assumptions 1 and 2, if the balanced growth rate g∗ exists, itsatisfies (27) and is unique and strictly positive.
Notice that Lemma 1 implies that the BGP is nondegenerate, along which si > 0 for
i = 1, 2.
Next, we show the existence and uniqueness of the relative price of outputs q ∗ on a
BGP:
Lemma 2. Under Assumption 1, the relative price of outputs on a balanced growth
path q∗ is uniquely determined and strictly positive.
Finally, we prove the existence of the uniquely determined balanced growth rate:
Lemma 3. Under Assumptions 1 and 2, the balanced growth rate g ∗ exists.

3.2.2

Existence and Uniqueness of Normalized Quantities

To obtain nondegenerate solution of the quantity variables, we establish:
Lemma 4. Under Assumptions 1 and 2, normalized capitals on a balanced growth path
are uniquely determined and strictly positive.
Once the BGP values of {xi , pj , g} are determined by Lemmas 1-4, {yj , c, wi /pi , vij }along the BGP can be pinned down in a recursive manner by
y = A−1 x; c = y1 − (g ∗ + δ)x1 ; wi /pi = si (g ∗ )/ρ; vii = β ii

yi /xi; vij = 1 − vii .
wi /pi(31)

This and Lemmas 1-4 imply,
Proposition 1. (Existence and Uniqueness of the BGP) Under Assumptions 1 and
2, a balanced growth path exists and is unique along which capital inputs, outputs andconsumption all grow at a common balanced growth rate g ∗ ∈ (0, ρ−1/(1−σ) −1) satisfying
Γ(g ∗ ) = 0, the balanced growth value of the relative price of outputs q∗ > 0 is pinned
down by f (q∗ ) = 0, and the balanced growth values of normalized quantities, factor
allocation shares and real factor prices, {xi , yj , c, vij , wi /pi } are all strictly positive,
determined by (25) and (31).

4Characterization of Local Dynamics

We now turn to characterizing the local dynamics of the recursive 4 × 4 dynamicalsystem of {x1t , x2t , p1t , p2t }, governed by the two fundamental equations (20) and (21)12around the BGP. It should be recognized that in this system that 4 × 4 dynamicalsystem describes three dimensional dynamics because of the linear homogeneity of thesystem in {x1t , x2t , p1t , p2t }, and one of the characteristic roots at the steady state isalways one as shown below. Thus, the dynamical system is locally determinate if there
exist at least two unstable roots and it is dynamically indeterminate if there exist atleast two stable roots.

4.1

Dynamical System

It is clearly seen from (20) and (21) that the price dynamics is determined by prices
alone. Moreover, as we will establish in the case with partial capital depreciation, theprice dynamics is related to the factor intensity ranking in the value or social sense.
b2 } (corresponding to theb1 , λThus, the associated characteristic roots are denoted by {λb matrix). Likewise, since the quantity dynamics is related to thedeterminant of the B
factor intensity ranking in the physical or private sense, the associated characteristicroots are denoted by {λ1 , λ2 } (corresponding to the determinant of the B matrix).
Using (21), we can characterize the price dynamics by,´−1(1 + g ∗ )σ ³
∂pt+1
I − Iδ + Â0
=
.
∂pt
ρ

(32)

b2 } solve (n = 1, 2):
b1 , λThe eigen roots evaluated at the balanced growth path {λ()∙³ ´−1 ¸−1
(1 + g ∗ )σbn I = 0,b0det
I − Iδ + A
−λ
ρ

or,

∙
³
´0 ¸
∗ σ
−1
bn I − Iδ + Ab
det (1 + g ) I − ρλ
= 0.

(33)

Comparing (33) and (24) with g = g∗ , it can be easily shown:

Lemma 5. Under Assumptions 1 and 2, one of the eigen roots governing the local
b1 = 1).
dynamics of the relative price of outputs is one (i.e., λb1 = 1 is a solution of (33) verifies our assertion above: the system exhibits
That λone-dimensional linear dependency because prices are in nominal terms.
With respect to the quantity dynamics, we can obtain from (20) the following:
¤
£
∂xt+1
= (1 + g∗ )−1 A−1 + (I − Iδ ) .
∂xt
13

(34)

The eigen roots evaluated at the balanced growth path {λ1 , λ2 } solve (n = 1, 2)£¤¡¢
det (1 + g ∗ )−1 A−1 + I − Iδ − λn I = 0.

(35)

b2 , λ1 , λ2 } and hence theAs we will establish in the next subsection, the values of {λproperty of local dynamics depend crucially on the rates of capital depreciation. This
contrasts sharply with the continuous-time endogenous growth models of Bond, Wang
and Yip (1996), Benhabib, Meng and Nishimura (2000), and Mino (2001).

4.2

Local Stability

In this section we demonstrate that, as long as capital stocks depreciate partially, i,e,0 < δ 1 , δ 2 < 1, the necessary and suﬃcient conditions for local determinacy and for
dynamic indeterminacy obtained in continuous time are neither necessary nor suﬃcientin discrete time.
To begin, we examine how the price dynamics are related to the sign of the socialb For illustrative purposes, we rule out the cases of degenerate
factor share matrix, ∆.
price paths (the first condition) and limit cycles (the second condition), which areb < 0:9
possible for ∆ihb +(1−δ 2 )βb ]
ρ[(1−δ 1 )β
21
12
∗ σ
b21 + (1 − δ2 ) βb12 .
Assumption 3. (1+g ∗ )σ 6=
and
(1+g
)
=
6
ρ
(1
−
δ
)
β
1
b
b +β
β
11

22

b < 0 (and hence
Consider then following inequalities which are valid only under ∆b22 < 1):b11 + ββihbbb21 + (1 − δ 2 ) β
b12 < (1 + g∗ )σ < ρ[(1−δ1 )β 21 +(1−δ2 )β 12 ] .
Condition P. ρ (1 − δ 1 ) β
b
b
β +β
11

22

We can then establish the following property concerning the price dynamics.

Lemma 6. Under Assumptions 1-3 with partial capital depreciation (i.e., 0 < δ 1 , δ 2 <1), the price dynamics possess the following properties:
b1 ;b2 < 1 = λb > 0, the price dynamics is stable with 0 < λ(i) for ∆b2 < −1 if and only if
b1 = 1 and λb < 0, the price dynamics is unstable with λ(ii) for ∆¯ ¯¯b ¯
b1 if and only if Condition P fails.
Condition P holds and stable with ¯λ2 ¯ < 1 = λ9Formally speaking, when the first expression holds for equality, it is a flip bifurcation point; when

the second expression holds for equality, it is a hopf bifurcation point.

14

We next turn to examining how the quantity dynamics are related to the sign ofthe private factor share matrix, ∆.
Lemma 7. Under Assumptions 1-3 with partial capital depreciation (i.e., 0 < δ 1 , δ 2 <1), the quantity dynamics possess the following properties:(i) for ∆ > 0, the quantity dynamics is unstable with |λ1 | > 1 and |λ2 | > 1;(ii) for ∆ < 0, there exists 0 < ρmin < 1, 0 < δ max < 1 and 0 < δ < 1 such that forany ρ ∈ (ρmin , 1), δi ∈ (0, δ max ), and |δ 1 − δ2 | < δ, the quantity dynamics features
a one-dimensional stable manifold with −1 < λ2 < 1 < λ1 .
An important message delivered by Lemmas 6 and 7 is that in contrast withb > 0 is not necessary for stable price adcontinuous-time endogenous growth models, ∆justments (as shown by Part (ii) of Lemma 6) whereas ∆ < 0 is not suﬃcient for stable
quantity adjustments (as shown by Part (ii) of Lemma 7). In the next two subsections,
we establish conditions under which local determinacy and dynamic indeterminacy,
respectively, emerge.
4.2.1

Local Determinacy

While consistent factor intensity rankings in the private and social sense need notlead to local determinacy within our discrete-time endogenous growth framework, we
consider a more special circumstance in which the production of the consumable gooduses consumable (physical) capital more intensively than the production of the pure(human) capital good in both value (social) and physical (private) sense, that is,b > 0.
Condition D. ∆ > 0 and ∆Proposition 2. (Local Determinacy under Partial Depreciation) Under Assumptions
1-3 with partial capital depreciation (i.e., 0 < δ 1 , δ 2 < 1), the dynamical system islocally determinate if Condition S holds.
It should be noted that since the proof applies to any dimensionality of N ≥ 2, thislocal determinacy results is also general to any multi-sector endogenous growth models
with socially constant-return sector-specific externalities.
In the continuous-time endogenous growth model, the necessary and suﬃcient condition for local determinacy is that the factor intensity rankings in the private and15b Our Condition S is stronger thansocial sense are consistent, i.e., sign(∆) = sign(∆).
the conventional requirement, though it is not necessary.
We next illustrate why Condition D is not necessary for local determinacy. Shouldb be negative, one may still establish:
both ∆ and ∆Proposition 3. (Local Determinacy under Partial Depreciation) Under Assumptions
1-3 with partial capital depreciation (i.e., 0 < δ 1 , δ 2 < 1), there exists 0 < ρmin < 1,
0 < δ max < 1 and 0 < δ < 1 such that for any ρ ∈ (ρmin , 1), δ i ∈ (0, δ max ), and|δ1 − δ 2 | < δ, the dynamical system is locally determinate if Condition P holds andCondition D fails.

Both Propositions 2 and 3 establish conditions for local determinacy. Under theconditions stated in Proposition 2, price adjustments are stable but quantity adjustments are unstable; under the conditions stated in Proposition 3, price adjustments are
unstable but quantity adjustments are stable. While Proposition 2 can be generalized
to any dimensionality of N ≥ 2, Proposition 3 cannot be.
Since Proposition 3 depends on not only the factor intensity ranking but the mabnitudes of time preference and capital depreciiateion rates. For example, Let {β 11 , β 21 ,b11 , βb12 } = {0.7, 0.75} so
β 12 , β 22 } = {0.5, 0.25, 0.6, 0.1} under which ∆ < 0, and {βb < 0. Further, take δ 1 = δ 2 = 0.1, σ = 1.05, ρ = 0.99 and α1 = α2 = 1. In thisthat ∆
case, the BGP is locally determinate.
Remark 2. It should be noted that if both capitals are fully depreciated in each period(δ 1 = δ 2 = 1), then the BGP is locally determinate, regardless of factor intensity
rankings. Specifically, in the case of full depreciation, from (33) and (35) the price andquantity dynamics respectively determined by the roots ofih£¤b = 0 and det (1 + g∗ )−1 A−1 − λI = 0.
b0 − λIdet ρ−1 (1 + g∗ )σ A
We can confirm that both roots of the first equation in the above are in the unite
circle, while those of the second equation are out of the unite cirle. Thus the BGP isalways locally determinate with full depreciation and partial depreciation is necessary
for indeterminacy to arise. To gain intuition, consider an alternative way to transformthe model into a stationary system using relative (rather than nominal prices), following the continuous-time analysis by Bond, Wang, and Yip (1996) and Mino (2001).
More specifically, using the second good as the numeràire, the relative price dynamics
under full depreciation are then governed by the following intertemporal no-arbitrage

16

condition,
qt
qt+1

=

(1 − δ 1 ) + w1,t+1 /qt+1
w1,t+1 /qt+1=,(1 − δ 2 ) + w2,t+1
w2,t+1

By the Stolper-Samuelson theorem, the sign of the responses of the righthand side of
the relative price dynamics depends crucially on the factor intensity ranking in theb > 0 and negative for ∆b < 0). Yet, changes in the relsocial sense (positive for ∆ative
in factor prices and, as a consequence,¯ changes¯price always induce magnified¯¯¯ dqt+1 ¯¯ dqt ¯
> 1, which implies ¯ dqt ¯< 1 and hence the price dynamics is¯ dqt+1 ¯
qt+1 =qt =q

qt+1 =qt =q

always stable. By similar arguments, under full depreciation, the two capitals become
essentially flow variables and the magnification eﬀects from the Rybczynski theorem
imply that the quantity dynamics are always unstable, reconfirming the local determinacy result.
4.2.2

Dynamic Indeterminacy

We next turn to studying the possibility of dynamic (local) indeterminacy. Consider the necessary and suﬃcient condition for dynamic indeterminacy obtained in
the continuous-time endogenous growth literature:b > 0.
Condition I. ∆ < 0 and ∆This condition states that the consumable good production uses pure (human) capital
more intensively in the private (physical) sense but uses consumable (physical) capitalintensive in the social (value) sense. We can now establish:
Proposition 4. (Dynamic Indeterminacy under Partial Depreciation) Under Assumption 1-3 and Condition I with partial capital depreciation (i.e., 0 < δ1 , δ 2 < 1), there
exists 0 < ρmin < 1, 0 < δ max < 1 and 0 < δ < 1 such that for any ρ ∈ (ρmin , 1),δ i ∈ (0, δ max ), and |δ 1 − δ 2 | < δ, the dynamical system is locally indeterminate.
Notably, Condition I is neither suﬃcient nor necessary for dynamic indeterminacy.
On the one hand, Condition I is not suﬃcient for dynamic indeterminacy, because
the magnitudes of time preference and capital depreciation rates both play essential
roles in addition to the factor intensity rankings. In particular, to generate dynamic
indeterminacy, we need both the time preference and the capital depreciation rates(ρ−1 − 1 and δ i ) to be suﬃciently small and the depreciation rates of the two capitals
to be suﬃciently alike. To be more concrete, consider {β 11 , β 21 , β 12 , β 22 } = {0.5, 0.25,
b12 } = {0.7, 0.65} so that ∆ < 0 and ∆b11 , βb > 0 and set δ 1 = δ 2 = 0.1,
0.6, 0.1} and {β17σ = 1.05, ρ = 0.99, α1 = 0.5 and α2 = 1. In this case, the dynamical system is locallyindeterminate (one-dimensional indeterminacy). Yet, by simply raising δ 2 to 0.2, thedynamical system becomes locally determinate.
On the other hand, Condition I is not necessary for dynamic indeterminacy.
Proposition 5. (Dynamic Indeterminacy under Partial Depreciation) Under Assumption 1-3 with partial capital depreciation (i.e., 0 < δ 1 , δ 2 < 1) and with Condition S
failing to hold, there exists 0 < ρmin < 1, 0 < δ max < 1 and 0 < δ < 1 such thatfor any ρ ∈ (ρmin , 1), δ i ∈ (0, δmax ), and |δ 1 − δ 2 | < δ, the dynamical system is locallyindeterminate if Condition P fails.
Again, to illustrate this possibility, consider the numerical example provided in theprevious subsection: by simply reducing α1 from 1 to 0.5, dynamic indeterminacyarises. Generally speaking, under the discrete-time setup, dynamic indeterminacy ismore likely to occur as a consequence of stable price dynamics (Lemma 6) but less so
as a result of unstable quantity dynamics (Lemma 7). Thus, Condition I turns out tobe neither necessary (as the conditions for stable price adjustments become weaker)
nor suﬃcient (as the conditions for stable quantity adjustments become stronger) for
dynamic indeterminacy.
One may compare this latter result with that in the discrete-time exogenous growthframework developed by Benhabib, Nishimura and Venditti (2002). Let us relabel
the consumption good sector in their paper as the consumable good sector and theinvestment good sector in their paper as the pure capital good sector. Then, they
establish the conditions for dynamic indeterminacy (in our notation with g∗ = δ i = 0):
b12 /2. Although our conditions are very diﬀerent, both suggestb > −β∆ < β 11 /ρ and ∆that even when Condition I is violated, dynamic indeterminacy may still emerge. Yet,
while Condition I is suﬃcient for dynamic indeterminacy in the exogenous growth
setting, it is not so in our endogenous growth framework. This is due to the increased
possibility of unstable quantity adjustments under the sustained endogenous growth
setup.

5

Concluding Remarks

This paper has reexamined the property of local dynamics in a class of two-sector
endogenous growth models with socially constant-returns production technologies withpositive sector-specific externalities. Within this class of models in continuous time,18

two fundamental propositions have been established: (i) the balanced growth path
is locally determinate if and only if the factor intensity rankings in the private andsocial sense are consistent; and, (ii) dynamic indeterminacy emerges if and only if theconsumable good production uses pure (human) capital more intensively in the private(physical) sense but uses consumable (physical) capital intensive in the social (value)sense. This paper shows that in discrete time, both propositions fail to hold.
Our main finding is thaqt the dynamic property is no longer determined by thefactor intensity ranking exclusively. Under partial depreciation, local determinacy is
obtained if the consumable good production uses consumable capital more intensively
in both private and social sense. When the consumable good production uses pure
capital more intensively in both private and social sense, the balanced growth pathmay be locally determinate or indeterminate, depending on the underlying preference
and production parameters. When the consumable good production uses pure capital
more intensively in the private sense but uses consumable capital intensive in thesocial sense, dynamic indeterminacy arises only if the time preference and the capitaldepreciation rates are suﬃciently small and the depreciation rates of the two capitals
are not too diﬀerent. Similar conclusions hold if we replace sector-specific externalities
by distortionary factor taxes.
Along these lines, one may revisit the issue raised by Benhabib and Perli (1994)
that whether dynamic indeterminacy can arise in set of plausible parameters withmoderate external eﬀects. For example, if we select {β 11 , β 21 , β 12 , β 22 } = {0.3, 0.55,
b12 } = {0.4, 0.35}, δ 1 = δ 2 = 0.05, σ = 1.5, ρ = 0.97, α1 = 0.02
b11 , β0.25, 0.4}, {βand α2 = 1, the resulting real interest rate, balanced growth rate, and share of pure(human) capital devoted to final goods production are 5%, 1.6% and 74%, respectively,
fitting with the U.S. data (though it should be noted that the selection of parameters is
not unique). In this case, the dynamical system is locally determinate. Moreover, one
may also find sets of parameters under which cycles may occur. Consider: {β 11 , β 21 ,b11 , βb12 } = {0.7, 0.75}, δ 1 = δ 2 = 0.1, σ = 1.05,
β 12 , β 22 } = {0.2091, 0.25, 0.6, 0.1}, {βρ = 0.99, α1 = 0.5 and α2 = 1. In this case, one of the roots governing the quantity
dynamics is zero and limit cycles arise. Of course, sector-specific externalities in themodel economy must be the driving forces of cycles. Yet, under which circumstances
cycles are likely to be present remains unexplored.10 Finally, it may be interesting10

b < 0. In the numerical example above,
As shown in Section 4.3, limit cycles are possible only for ∆b < 0. It is possible to find cases exhibiting limit cycles with ∆ > 0. Thus, thewe have ∆ < 0 and ∆19

to characterize the property of local dynamics using a more general class of sociallyconstant-return production technologies:¡¢
Yjt = αj Ψj v1jt X1t , v2jt X2t Gj (v1jt X1t , v2jt X2t ) ,where both Ψj and Gj are strictly increasing, strictly concave and twice continuously
diﬀerentiable and Ψj Gj exhibits constant-returns-to-scale. These are but three of many
possible avenues for future research.
consistency/inconsistency of the factor intensity rankings is not necessary for limit cycles to emerge.

20Appendix
Proof of Lemma 1: Define,bβ12Γ (g) ≡ (s1 (g))

bβ21(s2 (g))bβ12− (ρw1 )bβ21(ρw2 )

From (30), if a balanced growth rate g ∗ exists, it must satisfy Γ (g ∗ ) = 0. Utilizing the2P
bij = 1, we can see that,definitions of si and wi (i = 1, 2) and βi=1³´βb 12 ³´βb 21bbbbββ
11 β 2112 β 22,
Γ (g) = [(1 + g) − ρ (1 − δ 1 )] [(1 + g) − ρ (1 − δ 2 )] − ρα1 β 11 β 21
ρα2 β 12 β 22¢¡
where Γ (g) is increasing in g and, by Assumption 2, Γ (0) < 0 and Γ ρ−1/(1−σ) − 1 >0. Thus a solution g∗ satisfying Γ (g ∗ ) = 0 exists and is unique with 0 < g∗ <ρ−1/(1−σ) − 1. ¥
b
β
12

σ

b
β
21

σ

Proof of Lemma 2: Defining the diﬀerence of the left hand sides of (30) as f (q), we
can use the definition of si to get:
i
h
b /∆
b /∆
b
b
−β
β
21
12
− w2 q+ (δ 1 − δ 2 ) .
f (q) ≡ ρ w1 q

b > 0, f is an increasing function with limq−→0 f (q) = −∞ and limq−→∞ f (q) = ∞;If ∆b < 0, f is a decreasing function with limq−→0 f (q) = ∞ and limq−→∞ f (q) = −∞.
if ∆Hence, f (q) = 0 has a unique solution q ∗ > 0. ¥Proof of Lemma 3: From f (q ∗ ) = 0 in the proof of Lemma 2,
b

b

b

b

w1 q β 21 /∆ + δ 1 = w2 q−β 12 /∆ + δ 2 .
By combining this with (30), g∗ is determined. ¥Proof of Lemma 4: The key is to prove that the matrix I − A (g∗ I + Iδ ) is invertibleand [I − A (g ∗ I + Iδ )]−1 in (25) is a positive matrix. From (24), we have,hib =ρ I −Ab (g ∗ I + Iδ ) p > 0.
[(1 + g ∗ )σ − ρ (1 + g ∗ )] Ap(36)b (g ∗ I + Iδ ). Then (36) implies 0 < μLet μb be the Frobenius root of Ab < 1. Since∗b
0 < A < A, the Frobenius root μ of A (g I + Iδ ) satisfies 0 < μ < μb < 1. Thus,−1∗[I − A (g I + Iδ )] exists and is a positive matrix. Hence, x is uniquely determinedby (25) and x > 0. ¥Proof of Lemma 6: First, we consider the eigen roots of (33) governing the priceb1 = 1, the other root λb2 is given by,
dynamics. Recalling from Lemma 5 that λ#"µ³ ´−1 ¶−1σ−1∗0b2 = det ρ (1 + g ) I − Iδ + A
b
λ
−2

= ρ

¯
³ ´−1 ¯¯−1¯¯b0(1 + g ) ¯I − Iδ + A¯∗ 2σ ¯21

or, applying (9),
¯
¯−1
³ ´−1
¯
¯
−2
∗ 2σ ¯
0
−1 ¯
b
b
Ip ¯
λ2 = ρ (1 + g ) ¯I − Iδ + Iw B
⎡
⎤−1
−1
−1
−1
−1
b
b
b
b
1
−
δ
+
β
w
p
−
β
w
p
∆
∆
1
1 1
1 2
22
21
⎦ .
= ρ−2 (1 + g∗ )2σ det ⎣
−1
−1
−1
−1
b
b
b
b
−β 12 ∆ w2 p1
1 − δ2 + β 11 ∆ w2 p2

Utilizing (29),2P
bij = 1, and the definition of si , one can then obtain:
β

i=1

b2
λ

"Ã

!Ã
!
#−1
b21 s1 βb12 s2b22 s1b11 s2βββ
1 − δ1 +1 − δ2 +
−
b
b
b ρ∆
b
ρ∆
ρ∆
ρ∆
io−1
h
n
∗ 2σ b2bbb= (1 + g ) ∆ (1 − δ 1 ) (1 − δ 2 ) ρ ∆ + s1 s2 + ρ β 22 (1 − δ 2 ) s1 + β 11 (1 − δ 1 ) s2
io−1
h
n
∗ σ b
∗ σ
b
b
= (1 + g ) ∆ (1 + g ) − ρ (1 − δ 1 ) β 21 + (1 − δ 2 ) β 12.
(1 + g ∗ )2σ=ρ2Moreover, we have:

n
h
³
´i
h
io
1
1
∗ σ
b12 β
b21 + (1 − δ2 ) βb12b22 − βb21 − ρ (1 − δ 1 ) β
b11 β
(1
+
g
−1 =
)
1
−
β
b2
b
(1 + g ∗ )σ ∆
λ
n
³
´
h
io
1
∗ σ b
b
b
b
(1
+
g
β
−
ρ
(1
−
δ
)
+
β
)
β
+
(1
−
δ
)
β
=
1
2
12
21
21
12
b
(1 + g ∗ )σ ∆
´
³
1
b
b
=
β 21 s1 + β 12 s2 ,b
(1 + g ∗ )σ ∆

and,

h
³
´i
h
io
n
1
1
∗ σ
b11 β
b12 β
b21 + (1 − δ 2 ) βb12b22 − βb21 − ρ (1 − δ 1 ) β
1
+
β
+1 =
)
(1
+
g
b2
b
(1 + g ∗ )σ ∆
λ
³
´
h
io
n
1
∗ σ b
b
b
b
=
(1 + g ) β 11 + β 22 − ρ (1 − δ 1 ) β 21 + (1 − δ2 ) β 12 .
b
(1 + g ∗ )σ ∆b but sign( 1 + 1) depends both on sign(∆)
b and theHence, sign( λb1 − 1) = sign(∆),
b2
λ2magnitude of g ∗ , ρ and δ i .
b12 < 1. Thus, by Lemma 1,
b12 + βb > 0, β
For ∆
´
³
∗ σ
b
b
b
λ2 ∝ (1 + g ) − ρ (1 − δ 1 ) β 21 + (1 − δ2 ) β 12´³b21 + βb12
> (1 + g∗ )σ − ρ max {(1 − δ 1 ) , (1 − δ 2 )} β
> (1 + g∗ )σ − ρ max {(1 − δ 1 ) , (1 − δ 2 )}= min{s1 , s2 } > 0.
22

b2 < 1 and we establish: 0 < λb2 < 1 = λb1 under ∆b > 0.
Also, in this case, λb11 + βb22 < 1, we have to examine three possible cases.
b < 0 and hence β
For ∆First, consider the case where Condition P holds. In this case, one can easily see
∗ σ
b
that
h λ2 < −1 and hence thei price dynamics is unstable. Next, consider (1 + g ) <b2 < 1. Thisb21 + (1 − δ 2 ) β
b12 under which Condition P fails and 0 < λρ (1 − δ 1 ) β
implies that price adjustments are stable and monotonically converge to the BGP.
b +(1−δ 2 )βb ]
ρ[(1−δ1 )β21
12Finally, (1 + g∗ )σ >under which Condition P also fails, but nowbb
β 11 +β 22
b2 < 0 and hence price adjustments are stable and oscillate in converging to the−1 < λBGP.¥
Proof of Lemma 7: Recall that the quantity dynamics are governed by (35). Define,£¡¢¤F (λ) = det (1 + g∗ )−1 A−1 + I − Iδ − λI .

First, consider the case of ∆ > 0. We evaluate:£¤
F (1) = (1 + g∗ )−2 det A−1 − (g ∗ I + Iδ ) .

Using the transformed version of (5) along the BGP, aij = β ij pj wit−1 , and (29), we canderive:¤£F 0 (1) = − (1 + g ∗ )−1 (a11 + a22 ) |A|−1 − (2g∗ + δ1 + δ 2 )
¤
£¡
¢ −1
−1
− (2g∗ + δ 1 + δ2 )= − (1 + g ∗ )−1 β 11 p−1
2 w2 + β 22 p1 w1 ∆¤£= − (1 + g ∗ )−1 ρ−1 (s2 β 11 + s1 β 22 ) ∆−1 − ρ (2g∗ + δ1 + δ 2 ) .

Yet, from (25), [A−1 − (g ∗ I + Iδ )] x = z. Since ∆ > 0, [A−1 − (g ∗ I + Iδ )] has negative oﬀ-diagonal elements and positive diagonal elements, implying that the matrix,[A−1 − (g∗ I + Iδ )], has a dominant diagonal with every
principle minor being positive.
¤£
This results in F (1) > 0 and F 0 (1) = − (1 + g ∗ )−1 (a11 + a22 ) |A|−1 − (2g∗ + δ 1 + δ2 ) <0. Let d1 + d2 i be the roots of F (λ) = 0, i.e., F (λ) = (λ − d1 − id2 ) (λ − d1 + id2 ) = 0(where d2 = 0 means the roots are real). Then,
F 0 (1) = 2 (1 − d1 ) < 0.
Thus, the real part of two roots, d1 , are greater than 1 and F (λ) = 0 has two roots{λ1 , λ2 }, both outside the unit circle, |λ1 | > 1 and |λ2 | > 1.
Next, consider the case of ∆ < 0. Since det [I − A (g∗ I + Iδ )] > 0 and£¤det [1 − A (g∗ I + Iδ )] = |A| det A−1 − (g ∗ I + Iδ ) ,∆ < 0 implies |A| < 0 and hence det [A−1 − (g ∗ I + Iδ )] < 0 must hold true. This gives,£¤
F (1) = (1 + g ∗ )−2 det A−1 − (g∗ I + Iδ ) < 0.
¤£
Also (1 + g∗ )−1 (a11 + a22 ) |A|−1 − (2g + δ 1 + δ 2 ) < 0 implies F 0 (1) > 0. Thus, λ1 >1 and λ2 < 1. To see if λ2 is larger or less than −1, we evaluate F (λ) at λ = −1:£¤
F (−1) = (1 + g∗ )−2 det A−1 + I − Iδ + (1 + g ∗ ) I .
23

We rewrite det [A−1 + I − Iδ + (1 + g ∗ ) I] as follows:
½
¾
1
s1 s2
∗
∗
∗
∗
(2 + g − δ 1 ) (2 + g − δ 2 ) +[β 11 (2 + g − δ 1 ) s2 + β 22 (2 + g − δ 2 ) s1 ] +.
ρ∆ρLet assume that δ 1 = δ 2 = δ. Then s1 = s2 = s and the above expression becomes
∙
¸
1
s2
2
∗
∗
(2 + g − δ) +(β 11 + β 22 ) (2 + g − δ) s +.
ρ∆ρF (−1) > 0 for some s > 0 only if the following holds for some s > 0:
G (s) = s2 + ρ (β 11 + β 22 ) (2 + g∗ − δ) s + ρ2 (2 + g ∗ − δ)2 ∆ < 0.
Note that

G (0) = ρ2 (2 + g ∗ − δ)2 ∆ < 0.

There are two cases to consider. First, if σ > 1, then the following inequality holds aslong as g ∗ > 0.
(1 + g∗ )σ > ρ (1 + g∗ ) .
In order for g∗ > 0, s must satisfys > 1 − ρ (1 − δ) .
Hence, if ρ (1 − δ) is close to 1, then there exists some s > 1 − ρ (1 − δ) with G (s) < 0.
Second, if 0 < σ < 1, the transversality condition (1 + g ∗ )σ > ρ (1 + g ∗ ) is equivalent tos > ρ (g ∗ + δ). If g∗ and δ are both close to 0, then there exists some s > ρ (g ∗ + δ) suchthat G (s) < 0, where g∗ can be arbitrarily small as ρ is close to 1, from Assumption
2 and the definition Γ (g). By continuity, the same arguments apply for the case with
non-zero but small |δ 1 − δ 2 |, which completes the proof. ¥b1 . Moreb2 < 1 = λb > 0, Lemma 6 implies: 0 < λProof of Proposition 2 : Since ∆over, under ∆ > 0, Lemma 7 implies |λ1 | > 1 and |λ2 | > 1. The dynamical system
therefore has one root equal to 1, one stable root and two unstable root, implying local
determinacy. ¥b1 = 1 and λb2 < −1 whenb < 0, Lemma 6 implies λProof of Proposition 3 : Since ∆Condition P holds. Under ∆ < 0, Lemma 7 implies that there exists 0 < ρmin < 1,
0 < δ max < 1 and 0 < δ < 1 such that for any ρ ∈ (ρmin , 1), δ i ∈ (0, δ max ), and|δ1 − δ 2 | < δ, we have: −1 < λ2 < 1 < λ1 . Thus, the dynamical system has one root
equal to 1, one stable root and two unstable root, implying local determinacy. ¥Proof of Proposition 4 : From Lemma 6, the roots of the characteristic equation govb1 = 1 and 0 < λb2 < 1. By Lemma 7, there exists
erning the price dynamics are λ
0 < ρmin < 1, 0 < δ max < 1 and 0 < δ < 1 such that for any ρ ∈ (ρmin , 1), δi ∈ (0, δ max ),and |δ 1 − δ 2 | < δ, we have: −1 < λ2 < 1 < λ1 . The dynamical system therefore has
one root equal to 1, two stable root and one unstable root, implying one-dimensional
dynamic indeterminacy. ¥24Proof of Proposition 5 : From Lemma 6, the¯roots¯ of the characteristic equation gov¯b ¯
berning the price dynamics are λ1 = 1 and ¯λ2 ¯ < 1. Using Lemma 7, there exists

0 < ρmin < 1, 0 < δ max < 1 and 0 < δ < 1 such that for any ρ ∈ (ρmin , 1), δi ∈ (0, δ max ),and |δ 1 − δ 2 | < δ, one obtain: −1 < λ2 < 1 < λ1 . The dynamical system therefore has
one root equal to 1, two stable root and one unstable root, implying one-dimensional
dynamic indeterminacy. ¥25

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