Discussion Paper No. 827

CONFORMISM AND
STRUCTURAL CHANGE

Takeo Hori
Masako Ikefuji
Kazuo Mino

December 2011

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

Conformism and Structural Change∗
Takeo Hori†, Masako Ikefuji‡ and Kazuo Mino§
December 27, 2011

Abstract
We study structural change in a simple, two-sector endogenous growth model and showthat the presence of commodity-specific consumption externalities can be a source ofstructural change. When the degrees of consumption externalities are different between different goods, the two sectors grow at different rates, whereas the aggregate
economy exhibits balanced growth in the sense that capital stock and expenditure growat the same constant rate. Under the more restrictive condition such that the degrees ofconsumption externalities are the same, structural change does not occur. We also showthat the dependence of the benchmark consumption levels on the past consumption is
crucial for the divergent patterns of structural change across countries.
Keywords: Structural change, Consumption externalities, Two-sector growth model,
Kaldor facts
JEL classification numbers: E21, E30, O10, O41∗

We appreciate financial supports from the Joint Usage/Research Center Project of ISER from the Ministryof Education, Culture, Sports, Science and Technology. All remaining errors are ours.
†
Corresponding author: College of Economics, Aoyamagakuin University, 4-4-25, shibuya, shibuya-ku,
Tokyo, 150-8366, Japan. E-mail: t-hori@aoyamagakuin.jp.
‡
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047
Japan, e-mail: ikefuji@iser.osaka-u.ac.jp§
Institute of Economic Research, Kyoto University, Yoshida Honmachi, Sakyo-ku, Kyoto, 606-8501
Japan, e-mail: mino@kier.kyoto-u.ac.jp

1

1 Introduction
Seeking to answer the question of why there is so much evidence of large disparities instructural change and economic growth across countries is one of the central concerns ofgrowth economics. The existing literature on economic growth and development provides
some important insight into what factors drive structural change. Nevertheless, the previous
models are not completely consistent with observations of varying structural change as
discussed by Buera and Kaboski (2009).1 In this paper, we show that commodity-specificconsumption externalities can be a source of the diverse patterns of structural change using
a simple two-sector general equilibrium model.
Broadly speaking, two approaches have been used to study structural change. The
one is based on the assumption for production technology, while the other relies on nonhomothetic preferences assumption. In the studies concerning technological explanations,following Baumol’s (1967) seminal work, Ngai and Pissarides (2007) assume that thegrowth rates of total factor productivity (TFP) are different across sectors so as to derive
supply-induced structural change. They demonstrate that an aggregate balanced growth
path exists only if the utility function is logarithmic in the consumption composite consisting of one manufacturing good and an arbitrary number of consumption goods. Acemoglu and Guerrieri (2008) show that differences in factor proportions combined with capital deepening lead to an economy that features nonbalanced growth at the sectoral level,
and that aggregate growth is consistent with the well-known Kaldor facts suggesting approximate constancy of aggregate growth. In a framework based on demand-side reasons,
Kongsamut et al. (2001) employ a Stone-Geary utility function to illustrate the dynamic ofsectoral labor reallocation. However, in order for their growth model to be reconciled with
the Kaldor facts, a knife-edge condition is required. 2 In contrast, Foellmi and Zwimüller(2008) introduce a hierarchic utility function to obtain non-linear Engel-curves for the various products, which generates consumption cycles. Provided that demand functions only
depend on the relative position of the product in the hierarchy, they also show that structuralchange is consistent with the Kaldor facts.3
Unlike most previous works, we assume neither non-homothetic preferences of individuals nor sectoral differences in productivity growth. In fact, we assume that consumers’
preferences are homothetic in their private consumption and that each sector employs AK
technology of which TFP is constant. However, if commodity-specific consumption externalities are introduced, the utility function exhibits non-homotheticity from the social
perspective, which generates structural change reconciled with Kaldor facts. Under the as1

They find that traditional explanations are (1) unable to account for observed sectoral movement; (2)reliant on an unrealistic elasticity of substitution; and (3) inconsistent with the data between trends in sectoral
labor shares and trends in sectoral output shares.
2
The restriction requires that the market value of subsistence consumption of agricultural goods and thatof home production of services are the same.
3
In the case of an endogenous R&D framework, they show that hierarchic preferences stimulates the
incentive to innovate, which may generate multiple equilibria. Recently, Guilló et al. (2011) construct a
multisector overlapping generations model of endogenous technical change with the aim of showing thatbiased TFP growth and labor movement across sectors can be an endogenous response to the non-homothetic
preference.

2

sumption that the elasticity of substitution between the two goods is not equal unity, weshow that the economy converges to a balanced growth equilibrium when the same degreeof consumption externalities is applied commonly to both types of goods. However, when
the degrees of consumption externalities are different from each other, the output of one
sector grows faster than the other sector, whereas the aggregate economy exhibits balanced
growth. Moreover, we illustrate that the pattern of structural change depends on the initiallevels of benchmark consumption for given the initial capital stock in the latter case. Thus,
commodity-specific consumption externalities can drive structural change without sectorbiased technical change and non-homotheticity of private preferences. In addition, there
may exist various patterns in structural change.
The role of commodity-specific consumption externalities has been examined by someauthors such as Ravn et al. (2006), Doi and Mino (2008) and Hori (2011) who assume
that consumers put the same importance on all types of consumption of other households.
In such a situation, a consumer would care equally about consumption of bread and carof other consumers. However, the recent studies of the behavioral economics provide theevidence that consumers put the different importance on the different types of consumptionof other consumers.4 Our model generalizes to allow for differing degrees of consumption
externalities between different goods.
The rest of the paper is organized as follows: The next section sets up the model. Section 3 examines the evolution of the aggregate economy and shows the aggregate economy
reconcile with Kaldor facts. Section 4 examines the dynamics of the two sectors and describes an equilibrium. Section 5 analyzes the dynamic system for the case that the degrees
of consumption externalities of two goods are same and shows that structural change does
not occur. Section 6 analyzes the dynamic system and the stability of the equilibrium when
the degrees of consumption externalities of two goods are different and describes how structural change occurs. Concluding remarks are given in Section 7.

2 The Model
2.1

Firms

Time is continuous and denoted by t ∈ [0, ∞). We consider a competitive economy thathas two sectors. Sector 1 produces a good (called good 1), which can be either invested
for capital accumulation or consumed. Sector 2 supplies a pure consumption good (called
good 2).5 Good 1 is chosen as numeraire. The price of good 2 at time t is denoted by pt .
Firms in each sector are homogeneous. The number of firms in each sector is normalized
to one. The representative firm in Sector i (= 1, 2) produces good i by using the following

4

See Solnick and Hemenway (1998), Alpizar et al. (2005), and Carlsson et al. (2007), for example. They
investigate the positionality degree for different goods with empirical or experimental approaches.
5
Ngai and Pissarides (2007) consider the economy with m sectors where one of the m sectors produces
capital stock as well as consumption good, and the remaining m − 1 sectors produce only consumption goods.
The production side of our model is a special case of Ngai and Pissarides (2007) where m = 2 holds.

3

technology:
Yi,t = Ai Ki,tα (Bi,t li,t )1−α , 0 < α < 1,
where Yi,t , Ki,t , li,t and Ai > 0 are the output level, the capital input, the labor input andthe productivity in Sector i, respectively. It is assumed that the two sectors have the same
capital intensity. Bi,t represents the sector-specific externalities. Let us denote the average
capital and labor inputs of Sector i (= 1, 2) as K i,t and li,t , respectively. We have Ki,t = K i,tand li,t = li,t in equilibrium. We specify Bi,t ≡ K i,t /li,t so that the total production functionof the two sectors take simple AK-forms, Yi,t = Ai Ki,t , as in Carrol et al. (1997, 2000) who
study the role of consumption externalities in a one-sector growth model. The simplification
allows us to focus on the role of consumption externalities.
We assume that capital depreciation rate is equal to zero. Profit maximization in both
sectors yields:
A1
,
A2
K1,t
K2,t
wt = (1 − α)A1
= pt (1 − α)A2
.
l1,t
l2,t
rt = αA1 = pt αA2

and

pt =

(1a)
(1b)

The first line shows that due to our specification of the technology, the rental rate of capital,r, and the price of good 2, p, are both constant over time, which allows us to focus on therole of consumption externalities. In the following discussion, we omit the index t fromr and p. The second line gives the wage rate. The capital and labor market equilibrium
conditions are:K1,t + K2,t = Kt ,l1,t + l2,t = lt ,

(2a)
(2b)

where Kt and lt is the aggregate capital stock and labor supply. There is free capital andlabor mobility between the two sectors. The aggregate production of the economy is givenby:Yt ≡ Y1,t + pY2,t = A1 Kt .

2.2

(3)

Households

There is a continuum of identical households whose number is normalized to one. The
representative household consumes goods 1 and 2. The household inelastically supplies one
unit of labor, lt = 1. We assume that the instantaneous utility of the representative household
depends not only on her own consumption of the two goods but also on the benchmark
levels of consumption that are determined through outward-looking habit formation. The
instantaneous subutility of the representative household at time t is given by: hiε−θ2 ε−1 ε−1−θ1 ε−1,
if ε , 1, γ(c1,t h1,t ) ε + (1 − γ)(c2,t h2,t ) ε(4)
ut =  (c1,t h1,t −θ1 )γ (c2,t h2,t −θ2 )1−γ ,if ε = 1.
4

where ci,t is consumption of good i (= 1, 2) at time t, ε > 0 is the elasticity of substitution
between the two goods, and γ ∈ (0, 1) represents the importance of good 1. Here, hi,tdenotes the benchmark level of consumption of good i at time t. The presence of hi,t in(4) represents the commodity-specific consumption externalities, as in Ravn et al. (2006),
Doi and Mino (2008) and Hori (2011). As we discuss later, θi ∈ [0, 1) is associated with
commodity-specific consumption externalities. We assume that hi,t is a weighted sum of thepast average consumption of good i up to the time t:Z te−φi (t−u) ci,u du, φi > 0,
hi,t = φi−∞where ci,u is the average consumption of good i at time u in the economy at large. Differentiating the both sides of the above equation with respect to t yields:
ḣi,t = φi,t (ci,t − hi,t ).

(5)Concerning (4), the three points should be mentioned. First, when ε , 1 holds, ut canbe written as:
ε
 ε−1
 

!θ1  ε−1!θ2  ε−1

ε
ε 








c

  1−θ1 c1,t  + (1 − γ) c2,t 1−θ2 2,t  .
ut = 
γ c1,t




h1,t
h2,t


The subutility depends on the relative level of consumption, ci /hi , as well as the absolutelevel of consumption, ci .6 When both θ1 and θ2 are equal to zero, only the absolute levels of
consumption matter. If θ1 = θ2 = 1, only the relative level of consumption is all that matters.
For other values of θi (∈ (0, 1)), both are important. As in Carroll et al. (1997, 2000), we
assume that 0 ≤ θi < 1. When θi is strictly positive, an increase in hi,t negatively affects ut .
The preference of each household exhibits jealousy toward consumption of others. Because
the relative consumption of good i becomes more important as θi increases, we interpret θi
as an indicator of how much each household cares about the relative consumption of goodi. We call θi the degree of consumption externalities of good i. It is also to be noted that if
φi = +∞, then hi,t = ci,t holds so that external effects are only intratemporal.
The second relevant point is the degrees of consumption externalities. As discussed
in Introduction, the previous studies on the commodity-specific consumption externalities
assume that consumers put the same importance on the relative consumption of all good,which means θ1 = θ2 . However, the recent studies of the behavioral and experimental
economics suggest that the importance of the relative consumption varies depending on
characteristics of goods. Therefore, we consider the case of θ1 , θ2 as well as that ofθ1 = θ2 . Moreover, these studies typically find that people tend to care more about therelative consumption of the more visible goods like cars than that of the less visible good.
In our setting, we can reasonably consider good 1 that is used as capital good to be the more
visible good. So in what follow, the case of θ1 > θ2 is more relevant, although we also pay
attention to the case of θ1 ≤ θ2 .
θ1 γ 1−θ2θ2 1−γ1
When ε = 1 holds, ut can be written as ut = {c1−θ. Again, ut depends
1,t (c1,t /h1,t ) } {c2,t (c2,t /h2,t ) }on the relative level of consumption, ci /hi , as well as the absolute level of consumption, ci .
65The final point is concerned with homotheticity. Some authors such as Kongsamut etal. (2001) employ a non-homothetic utility function in order to construct a growth model
where outputs of different sectors grow at different rates. Apparently, (4) is homothetic fromthe perspective of private consumption, c1,t and c2,t . In the following discussion, we willobserve that even under a homothetic utility function, the presence of commodity-specificconsumption externalities gives rise to nonbalnaced growth.
The utility of the representative household is given by:Z ∞U=
e−ρt ln ut dt,(6)0

where ut is given by (4) and ρ > 0 is the subjective discount rate. To ensure a positive
growth rate in equilibrium, we assume αA1 > ρ.7 We use a logarithmic utility in (6) because
Ngai and Pissarides (2007) show that if the instantaneous utility function is a CES function,
structural change is consistent with balanced growth of the aggregate economy only when
the intertemporal elasticity of substitution is equal to one. The budget constraint is:ȧt = rat + wt − Et ,Et = c1,t + pc2,t ,

(7a)
(7b)

where at denotes the asset holdings of the household at time t that is equal to Kt in equilibrium. We solve the optimization problem of the household in two steps. In the first step, ut
is maximized subject to (7b), which yields:
!1−ε
!ε
c2,t
h2,t θ2
1−γ
.
(8)
=
c1,t
γp
h1,t θ1
Given h1,t and h2,t , (8) shows that the relative demand, c2,t /c1,t , is a decreasing function of
the relative price, p. Assume θ1 > 0 and θ2 > 0. Consider the case where the elasticity of
substitution is large so that ε > 1 holds. An increase in h1,t (h2,t ) reduces the relative demandfor c1,t (c2,t ). Therefore, in our specification, the preference of the household exhibits “running away from the Joneses (RAJ)” when ε > 1. When ε < 1 holds, an increase in h1,t (h2,t )has a positive effect on the relative demand for c1,t (c2,t ). Then, “catching up with the Joneses (CUJ)” prevails. Hence, a smaller ε represents a stronger degree of conformity among
consumers. The intuition is as follows: As h1,t increases, consumption of good 1 relative
to its benchmark, c1,t /h1,t , decreases, which has negative effects on ut . When the elasticity
of substitution is large, the household attempts to compensate the negative effects of h1,t by
substituting consumption of good 1 with that of good 2. Then, RAJ prevails. When theelasticity of substitution is small, the household attempts to compensate the negative effects
of h1,t by consuming good 1 more. Then, her preference exhibits CUJ. Most of the existingstudies in the literature assume CUJ, rather than RAJ. The case of ε < 1 is more related to
the literature although we also pay attention to the case of ε > 1.
From (7b) and (8), the following equations are derived:
c1,t =
1 + pz7Et θ 1−ε ≡ χt Et , and pc2,t = (1 − χt )Et .
h2,t 2
h1,t θ1

As we will see later, the growth rate of the economy is given by αA1 − ρ in a long-run equilibrium.

6(9)

where z ≡ 1−γ εγp

. Then, (4) can be written as:
ε

 ε−1
! ε−1

ε 

1
−
χ
ε−1

t
ut = Et γ(χt h1,t −θ1 ) ε + (1 − γ)h2,t −θ2 .
p

The second step of the household’s problem is to maximize (6) subject to (7a) and the above
equation. We then obtain:Ėt = (r − ρ)Et .
(10)
The transversality condition (TVC) is given by limt→∞ at e−rt = 0. Since at = Kt holds inequilibrium, TVC is satisfied if limt→∞ K̇t /Kt < r(= αA1 ).
Before closing this subsection, we derive the next two equations for later use.
!
ḣ1,t
ḣ2,t
ċ1,t ċ2,t−= (1 − ε) θ1
− θ2
,
(11)
c1,t c2,t
h1,t
h2,t
!
!
ċ1,t
c1,t
ḣ1,t
ḣ2,t
= αA1 − ρ + (1 − ε) 1 −
θ1
− θ2
.
(12)
c1,t
Et
h1,t
h2,t
We obtain the first equation by differentiating both sides of (8) with respect to t. The
derivation of (12) is presented in Appendix A. Equation (12) shows that when θ1 > 0 andθ2 hold, ḣ1,t /h1,t has a positive (negative) effect on ċ1,t /c1,t if ε < (>)1 holds while ḣ2,t /h2,t
has a negative (positive) effect on ċ1,t /c1,t . This is because when ε < (>)1 holds, CUJ (RAJ)
prevails.

2.3

Goods Market

Good 1 can be either invested for capital accumulation or consumed and good 2 is a pureconsumption good. Thus the goods market clearing conditions are:K̇t = A1 K1,t − c1,t ,
c2,t = A2 K2,t .
The above two equations, together with (1a) and (7b), imply that the aggregate capital
evolves according to:K̇t = A1 Kt − Et .
(13)

3 The Aggregate Economy
Before examining the dynamics of the two sectors, we examine the evolution of the aggregate variables, Kt and Et . Define ξt ≡ Et /Kt . From (10) and (13), we obtain:ξ̇t = {ξt − [(1 − α)A1 + ρ]}ξt .
Figure 1 shows the graph of the dynamics of ξ. The path that ξt converges to zero cannot be
equilibrium because K̇t /Kt converges to A1 and TVC is not satisfied. The path that ξt tends
7to +∞ is not equilibrium either because Kt converges to zero. Hence, the economy always
stay at ξ∗ ≡ (1 − α)A1 + ρ and then we have:
Et = [(1 − α)A1 + ρ]Kt ,(14)

for all t ≥ 0. In equilibrium, Kt and Et grow at the same constant rate, g∗ ≡ αA1 − ρ.
The capital income share remains constant at α = rKt /Yt . The interest rate also remains
constant. We then obtain the next proposition.
Proposition 1
In equilibrium, ξt remains constant at ξ∗ ≡ (1 − α)A1 + ρ over time. Kt and Et grow atg∗ ≡ αA1 − ρ. The capital income share and the interest rate remain constant.
Proposition 1 shows that the aggregate economy satisfies the facts pointed out by Kaldor.
Several points should be mentioned. Because of our specification of the technology, the interest rate and the growth rate of Kt always remain constant. If there are no sector-specificexternalities and then the production function takes a standard Cobb-Douglas form, the transitional dynamics arise and hence the interest rate and the growth rate of Kt no longer remain
constant over time. Note that the studies about structural change, including Acemoglu andGuerrieri (2008) and Kongsamut (2001), also focus on the steady state equilibrium where
the interest rate remains constant and capital stock grows at a constant rate. Therefore,
our analysis is in line with the literature. Second, as shown in Carrol et al. (1997, 2000),
when the benchmark level of consumption depends on the past consumption, the transitional dynamics arises even if the production technology takes an AK form. Because of thelogarithmic utility, the dynamics of the aggregate economy is independent of the benchmark
level of consumption and hence no transitional dynamics arises in our model. Finally, thefact that Kt and Et grows at the same constant rate does not necessarily mean that the two
sectors also grow at the same rate and there are no transitional dynamics of each production sector. In fact, the subsequent sections show that under reasonable conditions, the two
sectors grow at different rates and hence structural change occurs. Because consumption
externalities have no influence on the aggregate economy as mentioned just above, we can
examine the direct consequence of consumption externalities on structural change.
[Figure 1]

4 Dynamic System of the Two Sectors
To examine the dynamics of the two sectors, we define x1,t ≡ c1,t /Kt , x2,t ≡ pc2,t /Kt andηi,t ≡ hi,t /Kt (i =1, 2).
From (7b) and (14), we have:x1,t + x2,t = (1 − α)A1 + ρ.

8

(15)

If we use the definitions of xi,t and ηi,t and the fact that ci,t = ci,t holds in equilibrium, we
can derive the following three differential equations from (5), (12) and (14):
!"
!
!#
x1,t
x2,t
xt
ẋ1,t = (1 − ε) 1 −θ1 φ1− 1 − θ2 φ2− 1 xt ,(16a)(1 − α)A1 + ρ
η1,t
pη2,t
(16b)
η̇1,t = φ1 xt − (φ1 + g∗ ) η1,t ,φ2(16c)
η̇2,t = x2,t − (φ2 + g∗ )η2,t ,pwhere x2,t = (1 − α)A1 + ρ − x1,t and 0 ≤ x1,t ≤ (1 − α)A1 + ρ. η1,t and η2,t are state variables.
From the first equation of (9) and (14), we can derive:
x1,t =(1 − α)A1 + ρ θ 1−ε .
h 2
1 + pz h2,t θ1

(17)

1,t

This shows that the initial value of x1,t is determined by h1,0 and h2,0 . Hence, we should treat
x1,0 as a predetermined variable. Given x1,0 , η1,0 and η2,0 , when the degrees of consumption
externalities are different between goods 1 and 2 (θ1 , θ2 ), the dynamic system consists of(16a)-(16c).
When the degree of consumption externalities of good 1 is the same as that of good 2(θ1 = θ2 ≡ θ (0 ≤ θ < 1)), x1,t becomes a function of η1,t and η2,t from (17).
x1,t =(1 − α)A1 + ρ η θ(1−ε) .
1 + pz η2,t
1,t

(18)

Substituting (17) into (16b) and (16c) using (15), we can derive the dynamic system of η1,tand η2,t given η1,0 and η2,0 .
We next define a stationary equilibrium.
Definition
A stationary equilibrium (SE) is an equilibrium where ẋ1,t = η̇1,t = η̇2,t = 0 holds and c1,t ,
c2,t , h1,t and h2,t grow at constant rates.
It should be noted that in an SE, c1,t , c2,t , h1,t and h2,t do not necessarily grow at the samerate. An SE requires only that the grow rates of c1,t , c2,t , h1,t and h2,t are constant over time.
In the following discussion, we omit time index t from variables that are constant over time
when we analyze an SE.
When ε = 1 holds, the benchmark levels of consumption, h1,t and h2,t , do not affect thedemand for both goods 1 and 2 (see (8)). Consequently, consumption externalities have noinfluence on x1,t in equilibrium (see (16a), (17) and (18)). Because we are interested in theeffects of consumption externalities, we assume ε , 1 in the following discussion. Before
closing this section, we prove the next proposition concerning the existence of SE.
Proposition 2
Suppose ε , 1 and θ1 , θ2 . If there exists an SE, ċ1,t /c1,t , ċ2,t /c2,t must hold in the SE.
9(Proof) Suppose that there exists an SE where ċ1,t /c1,t = ċ2,t /c2,t holds. Then, (11) implies
θ1 ḣ1,t /h1,t = θ2 ḣ2,t /h2,t because of ε , 1. When ċ1,t /c1,t = ċ2,t /c2,t holds, ċ1,t /c1,t = ċ2,t /c2,t =g∗ (≡ Ėt /Et ) must hold so that (7b) is satisfied. Because c1,t and c2,t grows at the same rate as
Kt , x1,t and x2,t become constant. We denote those constants as x̂1 and x̂2 . From (16b) and(16c), we know that η1,t and η2,t converge to φ1 x̂1 /(φ1 + g∗ ) and φ2 x̂2 /{p(φ2 + g∗ )}, respectively. By definition of η1,t and η2,t , we must have limt→+∞ ḣ1,t /h1,t = g∗ = limt→+∞ ḣ1,t /h1,t ,which contradicts θ1 ḣ1,t /h1,t = θ2 ḣ2,t /h2,t because of θ1 , θ2 . Then, ċ1,t /c1,t , ċ2,t /c2,t must
hold in the SE. 
Proposition 2 tells that when θ1 , θ2 holds, there is possibility that the two sectors growat different rates and hence structural change occurs. In the following discussion, we firstconsider the case of θ1 = θ2 and show that structural change does not occur. After that, we
proceed to the case of θ1 , θ2 and observe that the presence of consumption externalities isa source of structural change.

5 The Case of θ1 = θ2 ≡ θ (0 ≤ θ < 1)
When θ1 = θ2 ≡ θ (0 ≤ θ < 1) holds, the dynamic system is given by (16b), (16c) and (18).
We first derive an SE. By setting η̇1,t = η̇2,t = 0 in (16b) and (16c), we obtain:
φ1 x1 = (φ1 + g∗ )η1 and φ2 x2 = p(φ2 + g∗ )η2 .

(19)

After we divide both sides of the first equation by those of the second one and rearrange theresulting equation using (15) and (18), we obtain:"# 1
η2
φ2 (φ1 + g∗ )z 1+θ(ε−1)
=
≡ ∆.
η1
φ1 (φ2 + g∗ )By use of (15), (18) and the above equation, we rearrange the two equations of (19) so as
to yield the SE values of η1 and η2 :
η∗1 ≡(1 − α)A1 + ρ
φ1(1 − α)A1 + ρ φ2 z∆θ(1−ε)
∗
,
and
η
≡
.
2
φ1 + g∗
1 + pz∆θ(1−ε)φ2 + g∗
1 + pz∆θ(1−ε)

Because ηi remains constant at η∗i that is strictly larger than zero, we have ḣ1,t /h1,t =ḣ2,t /h2,t = g∗ . From (18), x1,t also becomes constant and strictly positive. Hence, wefind that ċ1,t /c1,t = ċ2,t /c2,t = g∗ . Structural change does not occur in the SE we have justderived. Note that these results hold regardless of whether θ is equal to zero or not.
We next examine the transitional dynamics with the help of phase diagrams. From (18)and the first equation of (19), we can derive η̇1 = 0 locus:"
η2 = (pz)1
θ(ε−1)1# θ(1−ε)1+θ(ε−1)φ1 {(1 − α)A1 + ρ}
− η1
η1 θ(ε−1) .
∗φ1 + gNote that we have 1 + θ(ε − 1) > 1 − θ > 0 because of θ ∈ [0, 1) and ε > 0 (ε , 1). As
shown Panel (a) of Figure 2, when ε < 1 holds, η2 tends to +∞ as η1 approaches zero while10η2 tends to zero as η1 approaches φ1 [(1 − α)A1 + ρ]/(φ1 + g∗ ). In the region above (below)η̇1 = 0 locus, we have η̇1 < (>)0. When ε > 1 holds, η2 tends to zero as η1 approaches zero
while η2 tends to +∞ as η1 approaches φ1 [(1 − α)A1 + ρ]/(φ1 + g∗ ). In the region above(below) η̇1 = 0 locus, we have η̇1 > (<)0. From (15), (18) and the second equation of (19),
we can derive η̇2 = 0 locus:1"# θ(1−ε)1+θ(ε−1)
1
φ
{(1
−
α)A
+
ρ}
2
1
θ(ε−1) .
η1 = (pz) θ(1−ε)
η
−
η
2
2
p(φ2 + g∗ )As shown in Panel (a) of Figure 2, when ε < 1 holds, η1 tends to +∞ as η2 approaches
zero while η1 tends to zero as η2 approaches φ2 [(1 − α)A1 + ρ]/[p(φ2 + g∗ )]. In the region
above (below) η̇2 = 0 locus, we see that η̇2 < (>)0. When ε > 1 holds, η1 tends to zero asη2 approaches zero while η1 tends to +∞ as η2 approaches φ2 [(1 − α)A1 + ρ]/[p(φ2 + g∗ )].
In the region above (below) η̇2 = 0 locus, we have η̇2 < (>)0. Phase diagrams in Figure 2
show that the SE is stable and the economy converges to the SE.
[Figure 2]
We then obtain the following proposition.
Proposition 3
Suppose ε , 1 and θ1 = θ2 = θ ≥ 0. The economy converges to the SE where ċ1,t /c1,t =ċ2,t /c2,t = ḣ1,t /h1,t = ḣ2,t /h2,t = g∗ holds.
Proposition 3 holds regardless of whether θ is equal to zero or strictly positive. When θ1 =
θ2 holds, regardless of the presence of consumption externalities, growth is asymptoticallybalanced in the sense that the two sectors grow at the same asymptotic rate.
The contraposition of Proposition 2 tells that if there exists an SE where ċ1,t /c1,t =ċ2,t /c2,t holds, θ1 = θ2 holds in case of ε , 1. We obtain the next corollary from Propositions
2 and the results of this section.
CorollarySuppose ε , 1. There exist an SE where ċ1,t /c1,t = ċ2,t /c2,t if and only if θ1 = θ2 = θ(0 ≤ θ < 1) holds.
The next section shows that when the degrees of consumption externalities are different
between different goods (θ1 , θ2 ), the presence of consumption externalities generates
structural change.

6 The Case of θ1 , θ2
Again, let us assume that ε , 1. When θ1 , θ2 holds, the dynamic system is given by(16a)-(16c). In this case, there is no SE where c1,t and c2,t grow at the same rate as shown in
Proposition 2. However, the next proposition shows the existence of an SE where the two
sectors grow at the different (asymptotic) rates.
11

Proposition 4(i) Suppose (1 − ε)(θ1 − θ2 ) > 0. There exists an SE where the followings hold:ċ1,t ḣ1,t== αA1 − ρ ≡ g∗ ,c1,t h1,t
ċ2,t ḣ2,t K̇2,t 1 + (ε − 1)θ1 ∗===g ≡ ĝ∗ (0 < ĝ∗ < g∗ ),c2,t h2,t K2,t 1 + (ε − 1)θ2
φ1 [(1 − α)A1 + ρ]
x1 = (1 − α)A1 + ρ(≡ x̂∗ ), η1 =(≡ η̂∗1 ), η2 = 0(≡ η̂∗2 ),φ1 + g∗A2 k2,t φ2 + ĝ∗
c2,t
= lim
=
.
lim
t→∞ h2,tt→∞ η2,t
φ2

(20a)
(20b)
(20c)
(20d)

(ii) Suppose (1 − ε)(θ1 − θ2 ) < 0. There exists an SE where the followings hold:ċ2,t ḣ2,t K̇2,t=== αA1 − ρ ≡ g∗ ,c2,t h2,t K2,tċ1,t ḣ1,t 1 + (ε − 1)θ2 ∗ ˆ ∗==g ≡ ĝ (0 < ĝˆ ∗ < g∗ ),c1,t h1,t 1 + (ε − 1)θ1
φ2 [(1 − α)A1 + ρ] ˆ ∗x1 = 0(≡ x̂ˆ1∗ ), η1 = 0(≡ η̂ˆ ∗1 ), η2 =(≡ η̂2 ),p(φ2 + g∗ )c1,txt
φ1 + ĝˆ ∗
lim
= lim
=
.
t→∞ h1,tt→∞ η1,t
φ1

(21a)
(21b)
(21c)
(21d)

(Proof) See Appendix B
Proposition 3 provides many important implications. First, as shown in (20a) and (20b)(or (21a) and (21b)), when θ1 , θ2 holds, growth is nonbalanced in the sense that the two
sectors grow at the different (asymptotic) rates. We emphasize that because Proposition 1
holds, the aggregate economy exhibits balanced growth in the sense that Kt and Et grow as
the same rate.
In case of (1 − ε)(θ1 − θ2 ) > 0, both consumption and output of the two goods growat different rates in the SE. Output of Sector i(= 1, 2) grows at the same rate of Ki,t . Notethat x̂1∗ = (1 − α)A1 + ρ implies x2,t = 0 and K2,t /Kt = 0. Then, we have K1,t /Kt = 1
and hence K̇1,t /K1,t = g∗ > ĝ∗ = K̇2,t /K2,t holds in the SE. Output of Sector 1 grows at ahigher rate than that of Sector 2. When ε < 1 holds, the condition for nonbalanced growthis θ1 > θ2 . When CUJ prevails, the sector producing the more positional good grows faster.
Remember that when CUJ prevails, the growth rate of consumption of good 1 (ċ1,t /c1,t ) is
positively affected by the growth rate of the benchmark consumption of good 1 (ḣ1,t /h1,t ) andis negative affected by the growth rate of the benchmark consumption of good 2 (ḣ2,t /h2,t )(see (12)). The positive effect increases with θ1 while the negative effect increases with θ2 .
Consequently, when θ1 > θ2 holds, c1,t grows at a higher rate, which results in a higher
growth in Sector 1. Then, nonbalanced growth of the two sectors arises.
Also in case of (1 − ε)(θ1 − θ2 ) < 0, consumption of different goods grow at different
rates in the SE. Consumption of good 2 grows at a higher rate than that of good 1. From
12

the first equation of (21c), we know x2,t = (1 − α)A1 + ρ (= pc2,t /Kt = A1 K2,t /Kt ), which
implies K1,t /Kt = α − ρ/A1 > 0 and K2,t /Kt = 1 − α + ρ/A1 > 0. We then have K̇1,t /K1,t =g∗ = K̇2,t /K2,t . Outputs in the two sectors grow at the same rate in this equilibrium. Notethat Sector 2 does not produce capital good. To meet the demand for good 2, firms in Sector
2 must increase capital input in the production of good 2. Because capital good is produced
only in Sector 1, the demand of capital good for Sector 1 increases at the same rate as thedemand for good 2. Consequently, outputs in the two sectors grow at the same rate.
Which is the more realistic parameter restriction, (1 − ε)(θ1 − θ2 ) > 0 or (1 − ε)(θ1 − θ2 ) <0? As discussed in Subsection 2.2, we can reasonably consider good 1 that is used as capital
good to be the more visible and positional good, and hence the case of θ1 > θ2 is more
relevant. Further, CUJ prevails in case of ε < 1. The first case where (1 − ε)(θ1 − θ2 ) > 0
holds is more realistic. Under this realistic case, both consumption and output of the twogoods grow at different rates in the SE. Further, we know which sector grows at a higher
rate. Under the reasonable parameter restriction, θ1 > θ2 and ε < 1, our results predict that
consumption and output of the more positional good, good 1, grow at a higher rate thanthose of the less visible good.
Proposition 4 demonstrates that in case of (1 − ε)(θ1 − θ2 ) > 0, x2 (= (1 − α)A1 + ρ − x1 )tends to zero in the SE, which implies K2,t /Kt also tends to zero. However, this does notmean that Sector 2 does not produce anything because the economy approaches the SE only
asymptotically. At all points in time, Sector 2 produces positive amounts and grows at a
positive rate, ĝ∗ . At the same time, output of Sector 2 relative to that of Sector 1 decreases
because Sector 1 grow at a higher rate than Sector 2. In subsection 6.2, we examine thetransitional dynamics with the help of phase diagrams. There, it will be observed that incase of (1 − ε)(θ1 − θ2 ) > 0, shares of capital and labor allocated to Sector 2 gradually
decreases along the transitional dynamics and hence the structural change occurs. Before
we study the transitional dynamics, the next subsection examines the local stability of the
SE.

6.1

Stability

We first consider the case of (1 − ε)(θ1 − θ2 ) > 0. As in Acemoglu and Guerrieri (2008)
who also examine the stability of the SE of a two-sector growth model where output shareof a sector converge to zero, we linearize the dynamic system around the SE characterizedby (20a)-(20d).
 ẋ1,t  x1,t − x̂1∗ 



∗ 
 η̇1,t  = Ω  η1,t − η̂1  ,η̇2,t
η2,t − η̂∗2
where the Jacobian matrix, Ω, is given by: −(1 − ε)(θ1 g∗ − θ2 ĝ∗ ) + (1 − ε)θ2 (φ2 + ĝ∗ )
0


φ1
−(φ1 + g∗ )
Ω = 
−φ2 /p013(1−ε)pθ2(φ2 + ĝ∗ )2 φ2
0−(φ2 + g∗ )


 .



In deriving Ω, we use (20a)-(20d). Appendix C shows that the eigenvalues of Ω are givenby:µ1 = −(φ1 + g∗ )(< 0),µ2 = −(1 − ε)(θ1 g∗ − θ2 ĝ∗ )(< 0),µ3 = −{1 + (ε − 1)θ2 }φ2 − g∗ − (ε − 1)θ2 ĝ∗ (< 0).

(22a)
(22b)
(22c)

It is shown that µ2 and µ3 are both negative. In sum, the three eigenvalues of Ω are negative.
Remember that we should treat x1,0 as a predetermined variable and that η1 and η2 are also
non-jumpable variables. Because there are three stable roots, the SE is locally stable.
We next examine the stability of the SE in case of (1 − ε)(θ1 − θ2 ) < 0. Again, we
linearize the dynamic system around the SE characterized by (21a)-(21d).
 ẋ1,t  x1,t − x̂ˆ1∗ 



∗ 
 η̇1,t  = Γ  η1,t − η̂ˆ 1  ,η̇2,t
η2,t − η̂ˆ ∗2
where the Jacobian matrix, Γ, is given by: (1 − ε)(θ1 ĝˆ ∗ − θ2 g∗ ) + (1 − ε)θ2 (φ2 + ĝˆ ∗ )φ1
Γ = 
−φ2 /p(1−ε)θ1(φ1 + ĝ∗ )2
φ1−(φ1 + g∗ )

0



0


0
.
∗ 
−(φ2 + g ) 

In deriving Γ, we use (21a)-(21d). Appendix D shows that the eigenvalues of Γ are givenby:ζ1 = −(φ2 + g∗ )(< 0),ζ2 = (1 − ε)(θ1 ĝˆ ∗ − θ2 g∗ )(< 0),
∗

(23a)
(23b)
ˆ∗

ζ3 = −{1 + (ε − 1)θ1 }φ1 − g − (ε − 1)θ1 ĝ (< 0).

(23c)

It is shown that ζ2 and ζ3 are both negative. In sum, the three eigenvalues of Γ are negative,
so that the SE satisfies local stability. The next proposition summarizes the results obtainedso far.
Proposition 5
If ε , 1 and θ1 , θ2 , then the SEs we consider are locally stable.

6.2

Transitional Dynamics

Subsection 6.1 shows the stability of the SE. This section observes how structural change
progresses along the transition to the SE with the help of phase diagram. We focus on thecase where ε < 1 and θ1 > θ2 hold. For simplicity, we assume θ1 > 0 = θ2 .
The following three lines consist of ẋ1,t = 0 locus:
x1 = 0, x1 = (1 − α)A1 + ρ, η1 = x1 .
14

Note that 0 ≤ x1 ≤ (1 − α)A1 + ρ must hold from (15). Hence, we restrict our attention to
the region where this relationship holds. As shown in Figure 3, we have ẋ1 < (>)0 in theregion above (below) the graph of η1 = x1 . From (16b), we obtain η̇1 = 0 locus
η1 =

φ1
x1 .
φ1 + g∗

Because g∗ is strictly positive, the slope of η̇1 = 0 locus is less steeper than that of η1 = x1 .
We have η̇ < (>)0 in the region above (below) η̇1 = 0 locus.
The initial value of η1,t (≡ h1,t /Kt ) is determined by K0 and h1,0 . From (17), the initialvalue of x1,t is given by:
x1,0 =(1 − α)A1 + ρ(1 − α)A1 + ρ=.
θ1 (ε−1)
1 + pzη1,0 θ1 (ε−1) K0 θ1 (ε−1)
1 + pzh1,0(24)

Given K0 , both x1,0 and η1,0 increase with h1,0 . In Figure 3, which illustrates the phase
diagram in (x1 , η1 ) space, given K0 , we plot three initial points. Among three points, h1,0 issmallest in Point A and largest in Point C.
If the economy is initially located on Point A, the economy monotonically converges
to the SE. Along the transition, x1,t monotonically increases to its SE value. Note that
expenditure shares of good 1 and 2 are given by c1,t /Et = x1,t /{(1 − α)A1 + ρ} and pc2,t /Et =1− x1,t /{(1−α)A1 +ρ} and production shares of good 1 and 2 are given by Y1,t /Yt = αA1 −ρ+x1,t and Y2,t /Yt = (1 − α)A1 + ρ − x1,t . Along the transition to the SE, expenditure and output
shares of good 1 monotonically increase while those of good 2 monotonically decrease.
Because the production of Sector i is equal to Ai Ki (i = 1, 2), the share of capital allocatedto Sector 1 increases while the share of capital allocated to Sector 2 decreases along thetransition. From (1b), we have l2,t /l1,t = K2,t /K1,t , which indicates that labor allocated to
Sector 1 also increases while labor allocated to Sector 2 also decreases. Then, structuralchange steadily progresses along the transition. We emphasize that while structural change
progresses, the aggregate economy exhibits balanced growth in the sense that Kt and Et
grow at the same constant rate because Proposition 1 holds.
Figure 3 provides the further implication. From Proposition 1, we know that if there
are two countries that have the same technology and the same preference, the growth ratesof outputs in the two countries become exactly the same. If the initial values of Kt are
the same in addition, the output levels of the two countries also become exactly the sameover time and hence the two countries exhibit exactly the same performance of aggregate
economy. However, if the initial values of h1,t are different, the two countries follow thedifferent patterns of structural change. Consequently, the two countries have different expenditure and production shares of the two goods, and different shares of capital and labor
allocated to the two sectors at all points in time although these two countries have exactly
the same performance of the aggregate economy. Our model suggests that the presence ofconsumption externalities might be a source of the divergent patterns of structural change
across countries.
We finally show that the dependence of the benchmark consumption levels on the past
consumption is crucial for the divergent patterns of structural change across countries. Consider the case of φ1 = +∞ where the consumption externalities are only intratemporal. In
15equilibrium, we have h1,t = c1,t and η1,t = x1,t . From (12), (13) and (14), we obtain:nox1,tg∗ (1 − ε)θ1 1 − (1−α)A
1 +ρno x1,t .
ẋ1,t =x1,t
1 − (1 − ε)θ1 1 − (1−α)A1 +ρ
Figure 4 presents the graph of the above equation. (24) can be written as:x1,0 =(1 − α)A1 + ρ,
1 + pzx1,0 θ1 (ε−1) K0 θ1 (ε−1)

which shows that two countries endowed with the same level of K0 have the same initialvalue of x1,t . Then, from Figure 4, we know that in the case of φ1 = +∞, the two countries
that exhibit exactly the same performance of aggregate economy follow exactly the same
pattern of structural change.
[Figure 4]

7 Conclusion
This paper has demonstrated the important role of commodity-specific consumption externalities in generating structural change using a two-sector general equilibrium model.
Under the presence of commodity-specific consumption externalities, the utility function
exhibits non-homothetic from the perspective of social consumption, even though the individual’s preference is homothetic with respect to each good and production function of eachsector is of the AK form with constant productivity. We have shown not only that structuralchange occurs when the degrees of consumption externalities are different from each other,
but also that aggregate economy exhibits balanced growth in the sense that capital stockand expenditure grow at the same constant rate, which is consistent with the Kaldor facts.
Our results have shown the important implications for differences in country pattens ofstructural change. If the two countries have different initial benchmark levels of consumption goods, they follow the different patterns of structural change despite the same initialvalues of capital stock, technologies and preferences. Furthermore, their aggregate economy exhibit exactly the same performance. This could be a possible explanation for why
the patterns of structural change vary across structurally similar countries (Jorgenson andTimmer, 2011).
We have restricted ourselves to the case where the intertemporal elasticity of substitution in consumption is equal to one. A straightforward extension of the current model is torelax this restriction. The feature of the dynamics in our economy strongly depend on the
choice of the intertemporal elasticity of substitution in consumption. Also, the stability inan equilibrium could be affected by a change in the intertemporal elasticity of substitution
in consumption under the presence of commodity-specific consumption externalities.
As for another possible extensions of our investigation, we may consider a three-sector
model. In this paper, we focus on the role of commodity-specific consumption externalities
in generating structural change. We have shown that the two sector grow at the different
16rate in the equilibrium and either sector always grows faster than the other, depending onparameter values. It is interesting to pursue further studies on the hump-shaped share ofmanufacturing observed in the data. Employing a three-sector model would yield nonmonotonic reallocation of resources in the course of structural change.

Appendix
A The Derivation of (12)This appendix presents the derivations of (12). By differentiating (4) with respect to t, we
obtain:
(
!
!)
ċ1,t
ḣ1,t
ċ2,t
ḣ2,t
u̇t
1−ε
ε−1
ε−1
−θ
−θ
1
2
= ut ε γ(c1,t h1,t ) ε− θ1+ (1 − γ)(c2,t h2,t ) ε− θ1. (A.1)
ut
c1,t
h1,t
c2,t
h2,t
Using (4) and (11) in (A.1), we obtain:
ε−1

ḣ1,t
(1 − γ)(c2,t h2,t −θ2 ) εu̇t ċ1,t
=
− θ1
+ε
ε−1
ut c1,t
h1,t
uε

!
ḣ1,t
ḣ2,t
θ1
− θ2
.
h1,t
h2,t

(A.2)

ε−1

From (4), we have uc1,t = γu1/ε (c1,t h1,t −θ1 ) ε c1,t −1 where uc1,t is the partial derivative of uwith respect to c1,t . Differentiating both sides of this equation with respect to t yields:u̇c1,t u̇t 1 − ε u̇t 1 ċ1,tε − 1 ḣ1,t
− =
−
− θ1
.
uc1,t ut
ε ut ε c1,tε h1,t(A.3)

Using (7b) and the subutility function, we obtain:
uc1,t ∂ ln u ∂ ln u ∂E
1
=
=
= .
u
∂c1
∂E ∂c1 E
From the above equation and (10), we know that the left-hand side of (A.3) is equal toρ − αA1 . Therefore, from (A.2) and (A.3), we obtain:
!
ε−1
ċ1,t
(1 − γ)(c2,t h2,t −θ2 ) ε
ḣ1,t
ḣ2,t
= αA1 − ρ + (1 − ε)
θ1
− θ2
.
(A.4)
ε−1
c1,t
h1,t
h2,t
uε
We rearrange (8) as follows:
1 − γ c2,t h2,t −θ2γc1,t h1,t −θ1

! ε−1
ε
=

pc2,t
.
c1,t

By using (4) and (A.5), we have:(1 − γ)(c2,t h2,t −θ2 )
u

ε−1
ε

ε−1
ε

=

1−γ
γ

c2,t h2,t −θ2
c1,t h1,t −θ1


1 + 1−γ
γ

 ε−1
ε

c1,t
pc2,t
=1−
.
=
ε−1

c1,t + pc2,tEt
c2,t h2,t −θ2 εc1,t h1,t −θ1

By substituting these two equations into (A.4), we can derive (12).
17

(A.5)

B

Proof of Proposition 4

We first prove the case of (1 − ε)(θ1 − θ2 ) > 0. By substituting equations in (20c) into
(16a)–(16c), we obtain ẋ1,t = η̇1,t = η̇2,t = 0. Values in (20c) are the candidate of an SE.
Because ẋ1,t = η̇1,t = 0 and both x̂1∗ and η̂∗1 are strictly positive, we obtain (20a). Fromċ2,t /c2,t = ḣ2,t /h2,t , (11) and (20a), we obtain (20b). Because of 1 + (ε − 1)θi > 0 (i = 1, 2),
we have ĝ∗ > 0. The inequality (1 − ε)(θ1 − θ2 ) > 0 ensures ĝ∗ < g∗ , which is consistent
with η̂∗2 = 0. From (16c) and (20b), we obtain (20d).
We turn to the case of (1 − ε)(θ1 − θ2 ) < 0. By substituting equations in (21c) into
(16a)–(16c), we obtain ẋ1,t = η̇1,t = η̇2,t = 0. Values in (21c) are the candidate of an SE.
Because of x̂ˆ1∗ = 0, we have x2 = (1 − α)A1 + ρ > 0, which implies ċ2,t /c2,t = g∗ . In addition,
because η̇2,t = 0 and η̂ˆ ∗2 is strictly positive, we obtain (21a). From ċ1,t /c1,t = ḣ1,t /h1,t , (11)and (21a), we obtain (21b). Because of 1 + (ε − 1)θi > 0 (i = 1, 2), we have ĝˆ ∗ > 0. The
inequality (1 − ε)(θ1 − θ2 ) > 0 ensures ĝˆ ∗ < g∗ , which is consistent with η̂ˆ ∗1 = 0. From (16b)and (21b), we obtain (21d).

C The Eigenvalues of ΩOne of the eigenvalues of Ω is apparently µ1 = −(φ1 + g∗ ). The other two eigenvalues, µ2
and µ3 , satisfy:µ2 + µ3 = −(1 − ε)(θ1 g∗ − θ2 ĝ∗ ) − {1 + (ε − 1)θ2 }φ2 − g∗ − (ε − 1)θ2 ĝ∗ ,µ2 µ3 = (1 − ε)(θ1 g∗ − θ2 ĝ∗ )(φ2 + g∗ ) − (1 − ε)θ2 (φ2 + ĝ∗ )(g∗ − ĝ∗ )= [−(1 − ε)(θ1 g∗ − θ2 ĝ∗ )][−{1 + (ε − 1)θ2 }φ2 − g∗ − (ε − 1)θ2 ĝ∗ ].
Then, the eigenvalues of Ω are given by (22a)–(22c).
We next check the signs of µ2 and µ3 . By using the definition of ĝ∗ , we can rewrite µ2
as follows:(1 − ε)(θ1 − θ2 ) ∗g.
µ2 = −
1 + (ε − 1)θ2
We now consider the case of (1 − ε)(θ1 − θ2 ) > 0. And we have 1 + (ε − 1)θ2 > 1 − θ2 > 0
because of ε > 0 and 0 < θ2 < 1. Hence, µ2 is negative. We can show that µ3 is negative inthe following way:
µ3 < −{1 + (ε − 1)θ2 }(φ2 + ĝ∗ )< −(1 − θ2 )(φ2 + ĝ∗ )< 0.
The first inequality holds because of ĝ∗ < g∗ . The inequalities ε > 0 and θ2 > 0 ensure thesecond inequality. The last inequality holds because of θ2 < 1.

18

D The Eigenvalues of ΓOne of the eigenvalues of Γ is apparently ζ1 = −(φ2 + g∗ ). The other two eigenvalues, ζ2and ζ3 , satisfy:ζ2 + ζ3 = (1 − ε)(θ1 ĝˆ ∗ − θ2 g∗ ) − {1 + (ε − 1)θ1 }φ1 − g∗ − (ε − 1)θ1 ĝˆ ∗ ,ζ2 ζ3 = −(1 − ε)(θ1 ĝˆ ∗ − θ2 g∗ )(φ1 + g∗ ) + (1 − ε)θ1 (φ1 + ĝˆ ∗ )(ĝˆ ∗ − g∗ )= [(1 − ε)(θ1 ĝˆ ∗ − θ2 g∗ )][−θ2 g∗ ) − {1 + (ε − 1)θ1 }φ1 − g∗ − (ε − 1)θ1 ĝˆ ∗ ].
Then, the eigenvalues of Γ are given by (23a)–(23c).
We next check the signs of ζ2 and ζ3 . By using the definition of ĝˆ ∗ , we can rewrite ζ2 asfollows:(1 − ε)(θ1 − θ2 ) ∗ζ2 =g.
1 + (ε − 1)θ2
Now consider the case of (1 − ε)(θ1 − θ2 ) < 0. And we have 1 + (ε − 1)θ2 > 1 − θ2 > 0
because of ε > 0 and 0 < θ2 < 1. Hence, ζ2 is negative. It is also shown that ζ3 < 0 because
ζ3 < −{1 + (ε − 1)θ1 }(φ1 + ĝˆ ∗ )< −(1 − θ1 )(φ1 + ĝˆ ∗ )< 0.
The first inequality holds because of ĝ∗ < g∗ . The inequalities ε > 0 and θ2 > 0 ensure thesecond inequality. The last inequality holds because of θ2 < 1.

19

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ξ̇t

0

ξ∗

Figure 1. Dynamics of ξ22

ξ(a) ε < 1η2η̇1 = 0 locus

φ2 {(1−α)A1 +ρ}p(φ2 +g∗ )η∗2η̇2 = 0 locus

0

φ1 {(1−α)A1 +ρ}
φ1 +g∗

η∗1η1(b) ε > 1
η2
η̇1 = 0 locus
φ2 {(1−α)A1 +ρ}p(φ2 +g∗ )η∗2η̇2 = 0 locus0η∗1

Figure 2. Phase Diagrams: θ1 = θ2

23

φ1 {(1−α)A1 +ρ}
φ1 +g∗

η1

ẋ1 = 0 locus

η1Cη̇1 = 0 locus
BA(1 − α)A1 + ρ x1

0

Figure 3. Phase Diagrams: ε < 1 and θ1 > 0 = θ224ẋ1 = 0x0(1 − α)A1 + ρ x1

0

Figure 4. Phase Diagrams: ε < 1, θ1 > 0 = θ2 and φ1 = +∞25