

Department of Social Systems and Management
Discussion Paper Series

No.1297

Dynamical Analysis of the R&D-based Growth Modelwith a Regime Switch
byShiro KUWAHARA

July 2012

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPAN

Dynamical Analysis of the R&D-based Growth
Model with a Regime Switch
Shiro Kuwahara∗
July 13, 2012

Abstract
Several empirical studies suggest that advanced economies experience a growth regime switch from factor accumulation to knowledge
accumulation. To investigate the mechanism of such a regime switch,
this study develops a concise and flexible dynamic model based on
Romer (1990) by introducing two types of endogenously supplied R&D
input capital. The model replicates the growth patterns of developed
and underdeveloped nations, clarifies the important role that capital
plays in the difference between them, and presents several implications for interest-rate subsidies and official development assistance.
Further, it shows that if a country enjoying long-run growth has little initial capital, its initial economic development will be based on
capital accumulation. When the capital stock becomes sufficient for
supporting R&D, the economy will achieve long-run growth throughR&D.

Keywords: dynamics of regime switch, capital-accumulation-based growth,
R&D-based growth, effectiveness of economic policies
JEL Classification: E00, O00, O41

Address: Division of Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennodai Tsukuba, Ibaraki 305-8573, Japan. E-mail: kuwahara@sk.tsukuba.ac.jp. Phone/Fax:+81-29-853-6206
∗

1

1

Introduction

This study focuses on the mechanism of regime switch that underlies economic development. Since Solow (1957), total factor productivity (TFP)
growth and growth in human capital through R&D activities have been considered as a central factor in the determination of long-run growth. However, some studies imply that physical capital too plays an important rolein economic growth: Howitt and Aghion (1998) emphasize complementarity
between capital accumulation and long-run growth, and present some (indirect) evidence of this property. Howitt and Aghion (1998) construct a modelwith this complementarity, and Chandler (1990) demonstrates that the scale
expansion of enterprises generates R&D and product diversification. Furthermore, some empirical works show that the level of capital accumulation
affects the capability and/or profitability of R&D activities. Abramovitz andDavid (1973) show that America’s economic growth in the early nineteenth
century was more dependent on capital accumulation than on TFP growth.
Hayami and Ogasawara (1999) report similar results from Japanese pre-war
data. These works reveal that economies in the early stages of development
grow as a result of capital accumulation, but they subsequently pursue a
growth regime that is driven by R&D. On these perspectives, some empirical works report that newly industrializing countries in Asia grow mainly
through capital accumulation (as reported by Krugman (1994) and Young(1995)), but as Matsuyama (1999) discusses, these results are not itself negative for long-run growth. The important question is whether economies canmake a regime switch.
Thus, the main purpose of the present study is to analyze the dynamical properties of economic development with a regime switch, and as such,
we construct a model that directly relates R&D activities and capital accumulation by assuming that both human capital and physical capital are
accumulated and invested into R&D activities. We then replicate the dynamics of a regime switch: namely, we change the growth regime from capitalaccumulation-based growth to R&D-based growth.
In this regard, we extend Romer’s (1990) model that contains endogenouslydriven R&D activities, yields steady-growth and no-growth steady states, andgives a framework on optimal economic policies.1
1

Dynamical analysis and proof of stability were given by Arnold (2000).

2

Regime change and the realization of long-run growth have attracted attention in this decade because of their connection with the endogenous growththeory. Some theoretical works produce models for describing the regimechange from capital-based growth with decreasing returns to long-run positive growth (Zilibotti 1995; Matsuyama 1999; Funke and Strulik 2000; Galorand Moav 2004; Irmen 2005; Kuwahara 2007). The present study shares
its purpose and structure with those of Zilibotti (1995), Matsuyama (1999,
2001), Irmen (2005) and Kuwahara (2007), who focus on the regime switchfrom capital-based growth to R&D-based growth. While Zilibotti (1995) and
Irmen (2005) conduct an analysis by using the model of a competitive economy, Matsuyama (1999, 2001), Kuwahara (2007), and the present study use
models with monopoly power, which stems from the intellectual proprietary
right created by R&D activities. While Matsuyama (1999) and Kuwahara(2007) assume instantaneous monopoly power in the R&D sector and focus
on the multiplicities of the economic path,2 the present study focuses onlong-run growth with the regime change as a one-off, epoch-making event ineconomic development. In our study, the obtained growth path is a uniquesaddle-stable one, and no multiple equilibrium is obtained. However, thisuniqueness makes it possible to definitely analyze economic welfare.
Our main modification to Romer’s (1990) model is the consideration thatboth human capital and physical capital are used in R&D. This transformation of assumptions yields a regime switch in economic growth. The models of
Romer (1990) and this paper encompass two growth phases—growth without
R&D (capital accumulation phase) and growth based on R&D—and have theproperty that a sufficient stock of R&D input is necessary for growth based on
R&D. Romer (1990) assumes that R&D is executed by an exogenously given
human capital endowment,3 whereas we assume that endogenously supplied
capital is used for R&D activities; hence, the determination of whether or
Matsuyama (1999) focuses on the fluctuations between the two economic regimes—
capital-based and R&D-based growth—and Kuwahara (2007) concentrates on the multiple
steady states derived by expectations.
3
As early R&D-based growth models, see Romer (1990), Grossman and Helpman(1991), and Aghion and Howitt (1992), who assume that exogenously endowed humanresources and exogenously given R&D resources uniquely determine the long-run growth(or no growth). Furthermore, in these frameworks, if the R&D input factor grows, it yields
a drastic rise in the economic growth rate, as Jones (1995a, 1995b) critiques. In addition,some studies along this line conclude that introducing capital would not alter the essential
outcome (Grossman and Helpman 1991 Ch. 5; Aghion and Howitt 1998 Ch. 3).
2

3

R&D exists is endogenized. Because economic growth accumulates capital,
an economy with sufficient capital can shift from growth through only capitalstock to R&D-based growth. This is the brief sketch of the mechanism of theregime switch in our study.
Next, because we obtain a saddle stable path, we can derive optimal
policy rules in our model. As Jones (2002, p118) points out, resource allocation for R&D is distorted by three channels: (i) the “knowledge spillover”
effect that is caused by missing the market on R&D input for future contribution on improving productivity, (ii) the “stepping on toes” effect that isclassical externality, and (iii) the “consumer surplus” effect that stems frommonopoly pricing. Because our model shares properties, such as a distortion of intermediate goods pricing and freely used public knowledge capital,
with the Romer model, it causes a deviation in optimal resource allocationand thus the equilibrium capital stock to be excessively small, and becauseof a positive relationship between the GDP growth rate and capital stock,a lower GDP growth rate is engendered by a smaller capital stock. This
under-accumulation of capital can be eliminated by (only) interest subsidies. Whereas Romer (1990)’s policy analysis provides the result that thedetermination of the long-run growth regime depends on the endowment of
R&D input, we go one step ahead by endogenizing R&D input factor accumulation, and as a result, obtain the result that the determination of thelong-run growth regime is a deep parameter, such as R&D efficiency, rather
than an endowment. Therefore, our study concludes that any policy—that
adds factor endowment—has no effects on the long-run growth phase. Thus,
this optimal policy can enhance economic welfare, and affect the economic
growth rate. However, this optimal path for an economy with low efficiency
leads to a steady state with no growth path. In such a case, it is impossible
to set the economy on a long-run growth path through an optimal subsidy
policy. Namely, it is impossible to set such an economy toward long-run positive growth through extending knowledge, human or physical capital stock.
Thus, our study concludes that the effectiveness of official development assistance (ODA) is restrictive and conditional, which is also confirmed by someprevious empirical studies; for example, Boone (1996) asserts that foreign aid
does not increase growth rates in poor countries, and Burnside and Dollar(2000) report that the effectiveness of an aid policy is conditional.
The rest of the paper is organized as follows. Section 2 establishes themodel of a decentralized economy. The existence of the two types of steady4

states and their determinants is explained in Section 3. The dynamic property of the model is analyzed in Section 4, and Section 5 presents a welfare
analysis. Finally, Section 6 concludes the paper.

2

The Model

There is a competitively supplied final good which can either be consumed
as a consumption good (aggregate consumption is denoted by C) or invested
for physical or human capital accumulation (aggregate physical and humancapital stocks are denoted as K and H, respectively). Denoting final goods
production by Y , we have the resource constraint on final goods as Y =K̇ + Ḣ + C. Physical capital is used as an intermediate good for supplying
to the final goods sector (KY ) and for investment to create new intermediate
goods; that is, R&D (KA ). The market-clearing condition for capital imposes
K = KY + KA , where K is the amount of physical capital in the economy.
R&D is also assumed to be necessary for the input of human capital (H).
Both physical capital and human capital are accumulated by investing in final
goods. The economy is populated by infinitely lived representative agents; its
population is given by N and its growth rate is assumed to be exogenously
given at a constant n. One unit of population is assumed to inelasticallysupply one unit of labor; thus, aggregate labor supply can be identified withthe economy’s population scale. We assume that the economy is closed, final
goods are numéraire, and time is continuous.
The present study adopts a Romer-type production structure adopted in
Arnold (2000)—there are three primary factors of production: capital (K),labor (N ), and knowledge (A). Labor goes directly into the production of
final goods, and the capital used in production is first converted into a variety of differentiated and patented intermediate goods, which are indexedby i. The number of intermediate goods is denoted by A. These intermediate goods are aggregated by a symmetric CES, which in turn is combinedwith labor by means of a Cobb-Douglas technology. Thus, A denotes thescale of the cluster of the variety of intermediate goods, which gives the technological level in the economy, and A can consequently be regarded as thelevel of knowledge accumulation, or knowledge capital. Then, the production

5

function is specified as
Y =N

1−α

! A

x(i)α di,0 < α < 1,(1)0

where x(i) denotes the amount of intermediate good i, and [0, A] represents
the range of intermediate goods available in the current marketplace.
One unit of an intermediate good is assumed to be produced by η units ofcapital. Therefore, the capital allocated to the production of final goods KY"Ais quantified as KY ≡ 0 ηx(i)di. In a symmetric configuration regarding
intermediate goods, the equation x = x(i) converts the definition equation of
KY into KY = ηAx, or equivalently, x = (1/η)(KY /A). Since we assume that
final goods Y are consumed as consumption goods or invested in physical or
human capital accumulation, the following resource constraint for final goods
holds:(Y =)η −α N 1−α HYβ A1−α KYα = K̇ + Ḣ + C,(2)
where Ż denotes an increment in the value of Z.
The final goods sector is competitive, and Eq. (1) yields the first-order∂Yconditions (FOCs) of the production of final goods. These are given as ∂N=∂Yw and ∂x(i) = p(i), where w and p(i) denote the real wage and the price of
the ith sector intermediate goods, respectively. In this setting, the rental
price of human capital rH is equalized to be that of physical capital r if thehuman capital investment is executed.
In our model, all intermediate goods are protected by non-perishing patents
and the ith firm innovates the idea of ith intermediate good, holds the patent
for the ith intermediate good, and is a monopolistic supplier of the ith intermediate good. From the necessary condition that η units of capital are
required to produce one unit of intermediate goods, the profit of the firm
producing the ith intermediate good is given as Π(i) ≡ p(i)x(i) − rK ηx(i),where rK is the rental price of capital, and the firm producing the intermediate good maximizes this profit subject to the demand restriction∂Y= p(i). Under the symmetric configuration, this optimization condi∂x(i)1# 2 $ 1−ααtion yields the following: x(i) = x̄ = rK η
N, and p(i) = p = η αrK .
From Eq. (2) and the FOCs, wage rate w and interest rate r, and the profit
from the production of intermediate good Π̄ are, respectively, obtained asY
w = (1 − α) N
, rK = α2 KYY , and Π̄ = Π(i) = α(1 − α) YA , where, in equilibrium, the profit of each firm producing an intermediate good is equalized.
6

Therefore, Π̄ = Π(i).
R&D firms create new intermediate goods, and their patents stream perpetual monopoly profits Π̄, which comprise the revenue of R&D. Thus, theRτ"∞present value of this stream represents the value of R&D:V̄ (t) ≡ t Π̄(τ )e− t r(s)ds dτ .
Free entry of R&D is assumed. Therefore, if revenue from R&D exceeds its
costs, an infinite amount of capital would be allocated to R&D. Thus, revenue
from R&D cannot exceed the cost in equilibrium. If revenue from R&D does
not exceed its cost, investment in R&D is unprofitable, and no resources areallocated to R&D. In this case, an equilibrium without R&D (KA = H = 0)
occurs. Thus, if the economy is in equilibrium with positive R&D investment,
the revenues generated by R&D must be equated to its cost.
Since we assume that both human capital and physical capital are invested
to undertake R&D, firms that engage in R&D must pay a rental cost r for
their R&D activities in the process of innovation. Furthermore, innovation
is assumed to be the discovery of new intermediate goods that are added tothe existing set of intermediate goods; therefore, the expansion of the variety
of intermediate goods can be shown by the time derivation of knowledge
capital, Ȧ.
Finally, we assume that each R&D firm has one innovation; conversely,
each innovation has one patent. Therefore, the value of such a firm comprises
the “value” of one innovation. Then, the aggregate value in the macroeconomy is the summation of the values of all these firms, V̄ A; the aggregate
innovated value by R&D and its input cost are given as V̄ Ȧ and rKA +rH, respectively. Thus, the profit from R&D firm is given as Π̄R = ȦV̄ −r(KA +H)
and the free entry condition on R&D activities yields the following relationships between market equilibrium and capital allocation:%&%
Solow Regime: KA = H = 0>⇐⇒ r(KA + H)
V̄ Ȧ.
(3)
Romer Regime: KA , H > 0=
Whether an economy conducts R&D depends on condition (3). When KA >0, R&D occurs, causing the economy to grow through endogenous technological change. Following Matsuyama (1999), we term this regime as the
Romer regime. Condition (3) states that the equality rKA = Ṽ Ȧ holds inthe Romer regime. When KA = 0, no R&D occurs, and the economy grows
only by capital accumulation. Following Matsuyama (1999), we term thisregime as the Solow regime. In the Solow regime, the system yields theinequality r(KA + HA ) > Ṽ Ȧ for KA , H > 0.
7

On the Romer regime, differentiating Ṽ with respect to time provides thefollowing asset equation:˙rV̄ = Π̄ + V̄.
(4)Here, we introduce an R&D function—a function of increment of new variety(Ȧ) for the R&D input of physical and human capital (KA and H)—where
we assume an R&D input and that the function deceases with Ω(t), which
captures the increment in R&D difficulty. More specifically, we assume thefollowing R&D technology:Ȧ(t) = δA(t)KA (t)1−γ HA (t)γ,Ω(t)

0 < γ < 1,

δ > 0,(5)

where γ and δ denote the input efficiency share between physical and human capital and the total R&D efficiency parameter, respectively. Following
Romer (1990)’s setup, the R&D function linearly depends on existing knowledge capital A(t), and value of R&D is attributed to R&D input, (humancapital in the Romer model) with no R&D firm residual. Consequently, in thepresent model, where physical capital and human capital are R&D inputs,
we assume that the R&D function is homogeneous of degree one in humancapital and physical capital, which gives that value of R&D is attributed toboth human capital and physical capital with no R&D firm residual. Furthermore, Ω is introduced as in Segerstrom (1998). We need to impose theproperties of Ω as follows: (i) although we need not specify the entire path ofΩ, but the growth rate of Ω in the steady state must be determined (which
is done in the next section); (ii) rational agents in this economy are assumed
to know the entire path of Ω.
Combining Eqs. (4) and (5), the R&D firm’s profit Π̄R yields the FOCs
∂ Π̄
∂ Π̄
of the R&D firm as follows: ∂K= 0 and ∂H= 0. These conditions give theAγfollowing relationship on optimal R&D inputs: H = 1−γKA . Therefore, weγ
have the capital composition ratio on R&D as follows: ξ ≡ KHA = 1−γ. Thisis given as the rate of human capital against the physical capital devoted
to R&D, which is given by a deep parameter γ and therefore is a constant.
Using this relationship, the dynamics of A are given as
Ȧ = δA

KA ξ γ= δA(k̃ − k̃Y )ξ γ ,Ω

or gA = δ(k̃ − k̃Y )ξ γ ,(6)where k̃, k̃Y , and gA denote the R&D-efficiency-cost-unit-adjusted (henceforth, R&D-difficulty-adjusted) values of K and KY and the growth rate of8

A, respectively. More generally, z̃ denotes the R&D-difficulty unit value ofan aggregate variable Z; that is, z ≡ Z/Ω). We define the growth rate ofvariable Z as gZ ; that is, gZ ≡ Ż/Z.
Under the regime with positive R&D, using Eqs. (3) and (6) gives thevalue of R&D as follows:V γ
ξ = r(1 + ξ), and therefore δ ṽ Γ(γ) = r,(7)Ω"A
where V (≡ V̄ A = 0 Π(i)di) denotes the aggregate value of the R&D firm,ξγ
and ṽ ≡ V /Ω and Γ(γ) ≡ 1+ξ
= γ γ (1 − γ)1−γ . Because δ and γ are constantparameters and r is a constant in the steady state, (7) implies that V growsΠat the same rate as Ω in the steady state. Because we can write V = r−g,VY
where Π ≡ Π̄A and where Π = α(1 − α)Y is derived from Π̄ = (1 − α)α A , itis found that V grows at the same rate as Y at least in the steady state. We
specify this in the next section where steady state growth rates are derived.
To conclude the model, we examine the consumption decision of households. It is assumed that a representative household maximizes the utility
defined by! ∞c(t)1−θ − 1 −ρtU=e dt,
θ > 0,(8)1−θsubject to a budget constraint, where variable c denotes per capita value ofconsumption C, and deep parameters ρ and θ denote the subjective discountrate and constant relative risk aversion (CRRA), respectively. Furthermore,
we assume that this household also has a rational expectation. Namely, since
the economic path in this study includes a regime switch, the household accurately expects the period of regime switch and the entire sequence of factor
price on the super-long-run economic development process. Under these assumptions, we have an ordinary optimal condition regarding consumption
called the Keynes-Ramsey rule:
δ

θ

3

ċ(t)
= r(t) − ρ − n.
c(t)

(9)

Two Types of Steady States

We now analyze an economy in a steady state, wherein all variables growat constant rates. As shown below, our model contains two types of steadystates: one with R&D (and therefore, positive growth) and the other without
9

R&D (and therefore, no growth). We call these types of states the Romer
steady state (RSS) and the Solow steady state (SSS), respectively.
From the discussion in Section 2, we obtain the following. The first andsecond terms in (2) give the growth rate of Y as gY (t) = (1 − α)gA (t) +γ(1 − α)n + αgKY (t). HA = 1−γKA implies that the growth rates of physicalssss
capital and human capital are the same (namely, gK
= gK
= gK
= gH
),
Y
A
where s denotes the index on the value in the steady state. The second andssssthird terms in (2) imply that gYs = gK
= gK
= gH
= gH
= gCs must hold inYYa steady state. The discussion in the previous section implies that gΩ∗ = gY∗
is necessary for the existence of the steady state. Uniting these properties,s
we have that gYs = gK
= gCs = gAs + n = gΩs , or that gys = gks = gcs = gAs (≡ g s ).
For convenience, we assume that Ω(t) = A(t)N (t).
Substituting gcs = gAs into the Keynes-Ramsey rule given in Eqs. (9) and(6), we have
rs = θδ(k̃ s − k̃Ys )ξ γ + ρ + n.
(10)
Applying the notation of R&D-difficulty-adjusted value to Y , r, Π̃, anddynamics of V̄ , we obtain the following: ỹ(t) = η −α k̃Y (t)α+β , r(t) = α2 η −α k̃Y (t)α−1 ,'(˙π̃(t) = (1 − α)αη −α k̃Y (t)α , and r(t) − n ṽ(t) = π̃(t) + ṽ(t).
These notations
imply that ỹ and r can be considered as functions of k̃Y , and that taking Eq.
(7) into account, the dynamics of ṽ can be represented by the dynamics ofk̃Y .
Thus, in the steady state, the determination of the type of steady state
can be summarized as follows:%
Romer Steady States (RSS):⇐⇒ θδ(k̃ s − k̃Ys )ξ γ + ρ
Solow Steady States (SSS):&%(1 − α)δΓ(γ) s== r(k̃Ys ) − n
k̃Y .
(11)
α
>
In RSS, the condition given by (11) holds with equality, which yields thesteady state value of k̃Y and k̃ in RSS that are, respectively, denoted as k̃Y∗
and k̃ ∗ , where index ∗ presents a steady state value in RSS. k̃Y∗ and k̃ ∗ cannot
be analytically solved but the relationship between these two can be derived
as follows:k̃Y∗ =δ#δ θ ξ γ k̃ ∗ + ρθξ γ + (1−α)Γ(γ)α
10$ (≡ k̃Y∗ (k̃ ∗ )).

(12)

Thus, we obtain the relationship between steady state R&D-difficultyadjusted capital stock of both the aggregate value and that devoted to final
goods, k̃ and k̃Y , respectively, and the growth rate of the economy in RSS as
follows:4(1−α)δΓ(γ) ∗k̃ − ρ
∗
α
g =
.
(13)
(1−α)(1−γ)θ+α
The variable A denotes the variety of intermediate goods; after these varieties
are introduced, they never become trite. Consequently, the increment in A
is irreversible and g ∗ (= gA∗ ) cannot be negative. Thus, Eq. (13) immediately
gives the condition of positive growth ask̃ ∗ >)*αρ≡ k̃ .
(1 − α)δΓ(γ)(14)

Accordingly, the efficiency-unit capital stock in the steady state k̃ ∗ must belarger than the threshold k̃ to satisfy the feasible condition for RSS. As isshown later, this threshold of the R&D-efficiency-unit capital stock k̃ is also
derived in the necessary condition for SSS. Therefore, this threshold comprises the necessary condition on the determination of steady state (namely,
it determines whether the long-run growth regime is SSS or RSS). As isshown later, this condition also determines whether the Solow or Romertransitional regime is realized. As such, this value plays a central role inregime determination in this model, and we describe the properties of thisvalue here. The threshold value increases with ρ and decreases with δ, withthe efficiency parameter of human capital allocated to final goods productionαβ being indifferent to it. Since 1−α
is increasing with α, a higher α, which
results in lower labor share and higher capital share, reduces the likelihoodof RSS being realized. Furthermore, because Γ(γ) is symmetric U-shapedwith the lowest value at γ = 0.5, a more deviating efficiency of either capital(γ being nearer to 0 or 1) provides a smaller threshold value, which increases
the likelihood of RSS being realized.
We consider the mechanism for determining the steady-state capital stock∗k̃ . Substituting r = r(k̃Y ), k̃Y∗ = k̃Y∗ (k̃ ∗ ), and the definition of k̃ into the first
It is noteworthy that this growth rate closely resembles that of Eq. (13) in Romer(1990). One difference between the two is as follows. We have set the R&D input ascapital, which is endogenously accumulated. Therefore, (as derived below) we can endogenously derive the total endowment of capital, which is the input factor for R&D and forintermediate goods.
411and second terms in (11), we obtain the following condition that provides the
R&D-difficulty-adjusted per capita capital stock in the steady state:/0α−1γ
θ ξ γ ρ(k̃ − k̃)
δ
θξ
k̃
+
ρ
n+ρ+
= α2 η −α
k̃
.
(15)
γ
(δ θξ k̃ + ρ)
δ θξ γ k̃ + ρ
+
,. +
,.
≡L(k̃;α,γ,δ,ρ,n,θ)≡R(k̃;α,γ,δ,η,ρ,θ)By regarding k̃ as an argument of the two functions L(·) and R(·) defined in
condition (15), we can easily draw these two equations as a thin solid line anda thick solid curve, respectively, as shown in Figure 1. The R&D-difficultyadjusted capital stock in a steady growth path k̃ ∗ is uniquely determined by
the intersection of the two equations.
From Eq.(15) or Fig. 1, it is found that a larger population growthn yields a larger R&D-difficulty-adjusted knowledge stock in RSS (namely,dk̃ ∗ /dn > 0). Further, this result and Eq.(14) together yield that a larger
population growth n results in a larger growth rate in RSS (namely, dg ∗ /dn >0). Specifically, population growth is negatively related to economic growth.
This conclusion is the opposite of the findings of semi-endogenous growthmodels with labor R&D inputs. However, this is a desirable property of themodel in this study, because some empirical studies, such as Kelley (1988),
Kelley and Schmidt (1995), and Ahituv (2001), report a negative correlation
between population growth and economic growth. A rough sketch of thismechanism is given as follows. Condition (11) is the no-arbitrage condition
between the production of goods (the profit rate is given by the middle term)and R&D (given by the last term). Owing to the high population rate, alower kY is necessary for equating the middle and last terms in Eq. (11).
Combining this and Eq. (13), we get that a lower k̃Y leads to a lower growth
rate.
From Figure 1, the condition for k̃ ∗ to be a feasible solution—namely,(14)—holding is equivalent to L(k̃) < R(k̃) holding at k̃ = k̃, which yields
1+αα
the condition required in RSS: δ > Ψ(α, γ, η, ρ, n) where Ψ ≡ α− 1−α η 1−α (n +1ρρ) 1−α (1−α)Γ(γ).
Next, we now characterize SSS. In SSS, Eq. (11) holds with inequality,
and all capital is devoted to the production of final goods; therefore, k̃A∗∗ = 0.
That is, k̃ ∗∗ = k̃Y∗∗ , where ∗∗ denotes the steady state value in SSS. Therefore,

12substituting k̃ ∗∗ = k̃Y∗∗ into (11) yields the equilibrium capital stock in SSS:k̃∗∗= k̃Y∗∗ =1

α2 η −α
ρ+n

1
2 1−α

.

(16)

The inequality constituted by the second and third terms of Eq. (11) givesk̃Y ≤ k̃. Uniting this and the inequality in Eq. (16), we have the parameter condition for SSS as L(k̃) > R(k̃). Since this condition is mutually exclusive to that of RSS, the condition can also be written as follows:δ < Ψ(α, γ, η, ρ, n).
Note that the conditions for RSS and SSS obtained above are mutually
exclusive. Accordingly, the steady state is uniquely determined by the economy’s parameter set, and an economy with δ > Ψ has RSS, and that with
δ < Ψ does SSS. Thus, we summarize these results as follows.
Proposition 1. An economy has a unique long-run steady state—either
positive or no growth—that is determined by the following parameter condition:
&
%
&
>
RSS
δ
Ψ(α, γ, η, ρ, n) ⇔<SSS
That is, the parameter set {α, γ, η, n, ρ, δ} uniquely determines either of thesteady states.
Intuitively, these results imply that a country with higher R&D efficiency
and more patience realizes RSS, that is, long-run R&D-based growth, andthat a country with lower R&D efficiency and less patience realizes SSS,
that is, no-growth steady state. In contrast to the Romer model, initialendowment does not affect the long-run growth phase.

4

Transition Dynamics and Steady States

In this section, we analyze the transition dynamics. On the transition path,
we have two regimes, characterized as KA > 0 or KA = 0. We call these
the Romer regime and the Solow regime, respectively. We denote the initial R&D-difficulty-adjusted physical capital as k̃(0) = k̃0 . Furthermore,
we assume—for simplicity of discussion—zero initial endowment of human13capital (h̃(0) = 0).5

4.1

Local Transition Dynamics

A (decentralized) economic system comprises Eqs. (2), (4), (6), and (9).
This system is reconstructed as a system constituted by k̃, c̃, and k̃Y .
Substituting gc̃ + gΩ = gc + n(= gC ) and r = α2 η −α k̃Yα−1 in Eq. (9), we
obtain the dynamics of c̃ as follows:'(
˙ = 1 α2 η −α k̃Y (t)α−1 − ρ − n − σδ(k̃(t) − k̃Y (t))ξ γ c̃(t).
c̃(t)
σ

(17)

This dynamic equation is common to the two regimes. Regarding the dynamics of k̃ and k̃Y , each regime follows different dynamics as described
below.
4.1.1

Dynamics of the Economy in the Romer Regime

First, we examine the Romer regime. Here, we derive the dynamics of k̃ inthe Romer regime. From K̇(t) + Ḣ(t) = Y (t) − C(t), K(t) = KY (t) + KA (t),γ
H(t) = 1−γKA (t), the definition of k̃Y , and the assumption on Ω(t), we have
'
( '
(
˙
k̃(t)
= (1 − γ) ỹ(k̃Y (t)) − c̃(t) − n + δ(k̃(t) − k̃Y (t))ξ γ k̃(t)&%*k̃˙ Y (t) )+γ+ n + δ(k̃(t) − k̃Y (t))ξ γk̃Y (t).
k̃Y (t)(18)k̃Y (t))By using ṽ(t) = r(δΓ(γ)
and gṽ = r(t) − n − δΓ(γ)π̃(t), we obtain ther(t)dynamics of k̃Y asδΓ(γ)
nα2 η −αk̃˙ Y (t) =k̃Y (t)2 +k̃Y (t) −k̃Y (t)α .
α
1−α
1−α

(19)

Thus, Eqs. (17), (18), and (19) comprise the system in the Romer regime.
Because the dynamics of k̃Y guided by Eq. (19) are the function that
contains only k̃A as a variable, the dynamic properties of k̃Y are directly
obtained from Eq. (19). The dynamics of k̃Y around the steady state value
denoted by k̃Y∗ are found to be unstable; the phase diagram of k̃Y is given
See Appendix 1, which demonstrates that the case of h̃(0) > 0 has asymptoticallysame properties.
5

14

in Figure 2. Therefore, in order to realize RSS, it is necessary that k̃Y (t) =k̃Y∗ must be satisfied in, at least, the neighborhood of the steady state. Ifk̃Y (t) < k̃Y∗ , k̃Y (t) decreases following the dynamics of Eq. (19).
Thus, the dynamic system in the Romer regime must exist on the planek̃Y (t) = k̃Y∗ , which we term the Romer plane. Consequently, the system isreduced to a two-dimensional system comprising k̃ and c̃. Because the system
exhibits properties similar to the dynamic properties of the Ramsey model,
it can be easily verified that it has saddle stability.
4.1.2

Dynamics of the Economy in the Solow Regime

We now investigate the Solow regime where k̃A = 0 (which directly leads
to k̃(t) = k̃Y (t)) and where there is no demand for human capital. Thus,
the Solow regime also exists on the two-dimensional {c̃ − k̃} plane, which wecall the Solow plane. Under this condition, by using K̇(t) = Y (t) − C(t),K(t) = KY (t), and KA = H = 0, the dynamics of k̃ can be given as˙k̃(t)= α2 η −α k̃(t)α − c̃(t) − nk̃(t).

(20)

Under gA = 0, (17) can be written as'(
˙ = 1 α2 η −α k̃Y (t)α−1 − ρ − n c̃(t),c̃(t)σ

Thus, the dynamic system in this case is similar to that in the normal Ramseymodel. One difference is the interest rate, which reflects the monopoly pricing
in the intermediate goods sector, in which capital is one input and humancapital investment is the other. A Romer-type R&D-based growth model
contains distortional intermediate goods pricing, and our model assumes thatintermediate goods are produced using capital. Therefore, the interest rate(the capital rental price) is α times smaller than that in the normal Solowmodel. Other than the above, this system has the same properties as the
Solow model: it has a saddle stable steady state.

4.2

Global Transitional Dynamics and Steady States

Combining the local transition dynamics discussed in the previous subsection
and the steady state condition discussed in the previous section, we derive
the global dynamics in the present study. Because we have two types of15

steady states, RSS and SSS, we obtain two types of growth patterns related
to them. The phase diagrams related to these growth patterns are depicted
as Figures 3 and 4, respectively.
From Proposition 1, if an economy has RSS (that is, this economy satisfies δ > Ψ), then RSS is the economy’s unique steady state. In this case,if the economy has sufficiently large R&D-difficulty-adjusted capital stock—k̃(t) ≥ k̃Y∗ —the economy is on the Romer regime, and converges to RSS as
guided by the saddle stable path on the Romer plane (k̃Y = k̃Y∗ ), which isdrawn by the dotted region in Figure 3 (the transition path is drawn by abold line). If an economy has a sufficiently high R&D efficiency parameter,
but very small initial knowledge-adjusted capital stock—k̃(t) < k̃Y∗ —thenthe economy cannot start its growth process on the Romer regime because
the path that directly converges to RSS cannot be traced while satisfying the
no-arbitrage condition. Therefore, the economy rides on the Solow regime
and grows by only capital accumulation until the economy has capital stockk̃Y∗ . The rational expected transition path on the Solow plane in this caseis the path that connects to the saddle stable path on the Romer regime.
Saddle stability of the Solow regime implies that this is the unique paththat is consist with long-run feasibility, and this path always exist in thiscase.6 Thus, the assumption of initial capital stock derives the whole economic path with RSS, as is shown below. If an economy has efficient R&D(δ > Ψ) and a sufficiently small initial capital stock (k̃(0) < k̃Y∗ ), it grows
without R&D until it reaches k̃ = k̃, and then, starts R&D activities, growing with TFP. Therefore, it experiences a switch from the Solow regime to
the Romer regime, which is the regime switch described in the present study.
If the economy has efficient R&D (δ > Ψ) and a sufficiently large initial
capital stock (k̃(0) > k̃Y∗ ), it grows with R&D in the first stage of economic
development.
On the other hand, if an economy has lower R&D efficiency, and therefore,
is SSS (i.e., it satisfies δ < Ψ), Proposition 1 implies that SSS is the economy’s
unique steady state. In this case, the economy fails to initiate R&D even if
it contains more initial capital or knowledge stock than the threshold value(k̃(t) > k̃Y∗ ). This mechanism is explained as follows. Because the parametercondition δ < Ψ implies that the unique steady state of this economy is SSS,˙
It should be noted that the k̃ = 0 lines of both the Romer and Solow regimes intersect∗
at k̃ = k̃Y = k̃Y .
616

the equation in the condition (11)—derived from the arbitrage condition
between capital and the R&D firm—cannot hold after a finite period of timein the future. This expectation leads rational agents to refrain from investingin R&D. Thus, this economy grows without R&D and falls into SSS. This is
the mechanism underlying the no-growth trap in this study. Some moderately
developed economies fail to transit into long-run growth in the polarization
process; the mechanism presented here may explain the underlying cause.
The phase diagram of such an economy is depicted in Figure 4. In this case,for any initial capital endowment (k̃0 ∈ (0, ∞)), the economy converges to
SSS on the Solow plane. It should be noted that the intermediate goods arealso monopolized in this case, and that this system is a Ramsey model withimperfect competition. The distortion is summarized in the higher interest
rate: consumption growth rate is higher through the Keynes-Ramsey rule,and therefore the speed of convergence is high, but equilibrium capital stock
is lower through the decreasing return properties of capital.
Proposition 2. An economy has a unique steady state and a perfect-foresightsaddle-stable transition path that is convergent to the steady state. The longrun growth phase, showing either steady growth or poverty traps, is determined uniquely according to technological parameters (α, η, and δ) and preference parameter (ρ). The economy with RSS (and low initial capital endowment) experiences a regime switch from capital-accumulation-based growth
to R&D-based growth and realizes long-run growth. The economy with SSS
lacks the profitability for R&D investment and persistently stays in the Solow
regime, and is thus caught in poverty traps.

5

Optimal Growth and Economic Policy

The previous section shows the results of the present model with regard todecentralized economic growth. Following the perfect foresight path determined by the given parameters, an economy converges to a steady state witheither long-run growth or no growth. However, the present model contains
a distortion in intermediate goods pricing and an unfavorable equilibrium
termed as SSS. This section examines the possible roles that a government
plays in developing policies to promote economic welfare and development.

17

5.1

Command Economy

To obtain the welfare properties of a decentralized solution, consider thesocial planner formulation of the growth model treated above. A benevolent
government is assumed to maximize the representative household’s utility
function (8).
From the optimal conditions,7 we have the optimal capital allocation insteady state k̃Y∗op and the optimal growth rate of economy g ∗op , respectively,ask̃Yop =
∗

g =

δ

#

δ θ ξ γ k̃ op + ρθξ γ + (1−α)Γ(γ)α(1−α)δΓ(γ) opk̃ − ρ
α.
(1−α)(1−γ)θ+α$ (≡ k̃Yop (k̃ op )),

(21)

(22)

Eqs. (21) and (22) imply that in steady states, the determining relationshipbetween the economic growth rate and R&D-difficulty-adjusted capital in acommand economy accords with that in a decentralized economy.
From Eqs. (21) and (22), the condition of feasible positive growth g ∗ (k̃ op ) >0 is given as k̃ opSS > k̃. It should be noted that this condition is the sameas that in a decentralized economy. Thus, we obtain the following result.
In a command economy, the R&D-difficulty-adjusted capital stock, k̃ op∗ , inthe steady state must be larger than the threshold k̃ to satisfy the feasible
condition for RSS. This property is the same as that in a market economy.
This property implies that sufficient capital is necessary for steady growth.
Lacking this condition, no R&D investment (i.e., KA = gA = 0) is optimal.
For the no-growth case, R&D-difficulty-adjusted capital stock in a command
economy k̃ ∗∗op is given as follows. The poverty-trap R&D-difficulty-adjusted
per capita capital stock in a command economy k̃ ∗∗op is given as
k̃

∗∗op

1

αη −α
=
n+ρ

1
2 1−α
#

$
1
= α− 1−α k̃Y∗∗ > k̃Y∗∗ .

(23)

Further, we derive the R&D-difficulty-adjusted per capita capital stock forthe steady state under the case with positive R&D k̃ op∗ . The counterpart of
Eq. (15)—that is, the equation providing the steady-growth R&D-difficultyadjusted per capita capital stock in a command economy k̃ ∗op —is obtained
7

See Appendix A2 for detailed derivations.

18

as follows:θ ξ γ ρ(k̃ − k̃)n+ρ+= αη −αγ(δ θξ k̃ + ρ)+,. +≡L(k̃;α,γ,δ,ρ,n,θ)/δ θξ γ k̃ + ρk̃δ θξ γ k̃ + ρ
,-

0α−1

≡Rop (k̃;α,γ,δ,η,ρ,θ)

.

(24)

.

L(k̃; α, γ, δ, ρ, n, θ) is identical to the case of a market economy, and Rop (k̃; α, γ, η, ρ, θ)is α times the R in Eq. (15), R(k̃; α, γ, δ, η, ρ, θ). Rop is drawn in Figure 1 asa broken line. Because the difference between a decentralized economy anda command economy is that R is α times Rop , the properties of the steadystate are similar in both cases. In the steady state of an optimal economy,
lower ρ, η, and n lead to a higher per capita GDP growth rate. This is thesame as in the steady state in a decentralized economy. However, equilibrium
capital stock in a command economy is larger than that in a decentralized
economy. Therefore, the growth rate of this economy is higher than that ina decentralized economy.
Because L(k̃) and Rop (k̃) are monotonous increasing and decreasing functions, respectively, the condition k̃ opSS > k̃ is equivalent toL(k̃) < Rop (k̃).

(25)

Substituting k̃ into L(k̃) and R(k̃), we obtain the following inequality:δ > Ψop (n, η, ρ, α),

(26)

1
1−α

where Ψop ≡ α Ψ.
Summarizing the above discussions, we obtain the equilibrium capitalstock in a command economy k̃ op as follows:
Proposition 1’. An economy has a unique optimal long-run steady stateof positive growth or no-growth traps that is determined by the following condition:
&
%
∗op
op
k̃
if δ > Ψk̃ op =∗∗opk̃
if δ < Ψop&positive growth (RSS)⇒ Long-run optimal growth is. (27)
no growth (SSS)
1

Because 0 < α 1−α < 1 for 0 < α < 1, it should be noted that Ψop < Ψ,which implies that the necessary condition for RSS is less restrictive in acommand economy than in a decentralized economy.
19

5.2

Effects of Economic Policies

In this section, we introduce subsidies and taxes into our model and discuss
the effectiveness of economic policies. We propose that a constant rate subsidy s > 0 (or a tax if s < 0) is levied (or provided) for interest (rental priceof capital) and the profit of the intermediate sector, as below:rs ≡ (1 + sr )r = (1 + sr )α2

Y
,
KY

Y
π s ≡ (1 + sπ )π = (1 + sπ )α(1 − α) ,A
where sr and sπ represent the rate of interest and profit subsidies, respectively.
The existence of distortion in the intermediate goods market leads thedecentralized economy to accumulate less R&D-difficulty-adjusted capitalstock than a command economy. For this reason, policies that promote capital accumulation by subsidizing the interest rate always improve economic
welfare.
The government is assumed to finance these subsidies using lump-sum
tax revenues; total tax revenue is expressed as T LS . We assume that the
government maintains a balanced fiscal policy: the budget constraint sr K +sπ A = T LS is always satisfied. Translating r and π in Eqs. (17) and (19) to
rs and π s , we determine both dynamics after taxation (or subsidy) as follows:
(
1'
c̃˙ =
(1 + sr )α2 η −α k̃Yα−1 − ρ − n − σδ(k̃ − k̃Y ) c̃,σ2 −α˙k̃ = δ (1 + s )k̃ 2 + n k̃ − (1 + s ) α η k̃ α .
Y
π Y
Y
r
α
1−α
1−α Y

(28)
(29)

For optimal growth, Eqs. (28) and (29) must correspond with Eqs. (39) and(40) (in Appendix), respectively.
Thus, we obtain the result that optimal growth and capital allocation are
realized by the following subsidy policy:s∗r =1− 1 > 0,αs∗π = 0.

(30)

Because α ∈ (0, 1), s∗r is always constant and negative, implying that thispolicy is perennially effective and that it increases economic welfare.
The above part showed that a governmental policy of interest rate subsidies can increase the economy’s welfare. However, can an optimal subsidy
20

deliver an economy from a poverty trap onto steady growth? The answer, atleast partially, is affirmative.
To answer this question, we provide Table 1 that summarizes the resultsof the determination of steady states on market and command economies, derived in Propositions I and I’. For both market and command economies, thesteady states are determined by R&D efficiency δ and other deep parametersn, η, ρ, and α that are contained in Ψ and Ψop . This table implies that thenecessity of R&D efficiency for RSS is less restrictive in a command economy
than in a market economy, and as such, if a country has the parameter set
Ψop < δ < Ψ and a market economy without a subsidy, the country is stuck
in SSS although the optimal steady state of the economy is long-run positive growth (RSS). In this case, the country is capable of long-run positive
growth, but monopoly power exercised in the decentralized economy draws
the economy into a no-growth trap. In this situation, the optimal subsidy
policy transforms the economy’s steady state from SSS to RSS. In this manner, the economy with Ψop < δ < Ψ can shift from a poverty trap to long-run
growth through relevant subsidy policies. Thus, the necessity of R&D efficiency for RSS is less restrictive in a command economy than in a market
economy. As a result, an economy with middle-range R&D efficiency fails
to realize market-economy-driven R&D-based long-run growth, but relevant
subsidizing policies can enable it to experience long-run growth.
If an economy’s parameter set is δ < Ψop , its situation is worse. In thiscase, the economy has SSS as the optimal steady state for any stock variableendowments; therefore, convergence to a no-growth equilibrium (SSS) is the
economy’s welfare maximizing path, and consequently, even if the government provides relevant subsidies derived above, SSS stays. If this country
desires long-run growth, relevant parameters (for example, R&D efficiency δ)must be improved. Further, any ODA, such as licensing technology (which
implies an increment in knowledge A in our model), capital contribution(which implies an increment in capital stock K), and setting up of an educational institute (which tries human capital accumulation H), makes nochange in the determination factor of the steady state because the condition
given in Proposition I’ contains only deep parameters and does not endow human or physical capital stock or knowledge accumulation. Thus, the steadystates of the aided countries remain completely unaffected (if such accumulation has no external positive effects on the efficiency condition of R&D
through deep parameters). The only result of such ODA is a jump in the21same transition path. Consequently, the economy will fall into a no-growth
path in the long run by adjusting the stock of A or K. Therefore, for effective ODA, it is necessary to aid an economy to make structural changes that
improve deep parameters, for example, enhancing efficiency parameters suchas the cost of the production of intermediate goods η or R&D efficiency δ.
Thus, the above results can be summarized as follows.
Proposition 3. Social welfare increased with the interest-rate subsidy rates∗r which increases the long-run growth rate along a steady growth path. Moreover, the subsidy s∗r extricates an economy from poverty traps if its RSS pathis optimal (that is, if condition δ > Ψop is satisfied). Setting an economy with
SSS on a long-run growth path requires structural improvement. Technology
licensing or capital stock will not be effective for this purpose.

6

Conclusion

This study developed a model with endogenously-accumulated R&D inputs
and investigated the mechanics of the growth process with a regime switch.
It shows that the equilibrium (physical) capital stock is positively related
to the long-run growth rate. Capital stock depends negatively on the costof intermediate goods and the subjective discount rate, which determines
the steady state of the economy. Each economy has a unique steady state
and a unique transition path converging to it. To achieve long-run growth,a country must have high R&D efficiency, low cost of intermediate goods,
and a subjective discount rate. An economy that does not satisfy these
conditions languishes in a no-R&D regime and a poverty trap. If the economy
has low initial endowment of (R&D-difficulty-adjusted) capital and a steadystate with a long-run positive growth, a regime switch occurs. Low capitalendowment lowers R&D profitability on R&D, and hence, no R&D takes
place. Further, when capital stock becomes sufficient for supporting R&D,
the economy will achieve long-run growth through R&D.
Because our model incorporates a monopoly on the production of intermediate goods, the economy contains a distortion, and therefore, the economic
policy is effective. This distortion appears in the economy’s interest rate
and results in lesser long-run capital stock. Therefore, governmental policies for subsidizing the interest rate and augmenting steady-state capital can22steer the economy to an optimal path. Economic welfare can always be improved through this subsidization. However, under some parameter sets, theeconomy’s optimal path is a no-growth path. In this case, some policies for
guiding the economy toward long-run growth by increasing physical capitalor knowledge capital will not enhance economic welfare. Effective aid should
improve the economic environment for R&D by, for example, enhancing the
productivity of intermediate goods or the efficiency of R&D investment. A
patent is regarded as a written or a formal knowledge, whereas deep parameters such as R&D efficiency and the productivity of intermediate goods are
considered non-written or implicit knowledge. Future research should focus
on economic analysis and foreign aid in order to gain a deeper sight into this
topic.

7

Appendix

A1. Positive Initial Endowment of Human CapitalIn this appendix, we consider the case of H(0) = H0 > 0. If the profit rateof human capital is equated with physical capital, it is simply the Romer
regime, and as such, in this section, we consider the case wherein the profit
rate of human capital is smaller than that of physical capital. In this case,Ḣ = 0 and r > rH hold, and therefore, human capital endowment is fixedat the initial endowment H0 , and R&D is executed to hold the no-arbitrage
condition on capital profit rate r. Then rH is determined at the level thatsatisfies the zero-profit condition on R&D.
Therefore, the equilibrium condition on this regime is given by∂ΠξA−γ= (1 − γ)δ
V̄ − r = 0,
∂KA
N (t)

∂Π
ξA1−γ
=γδ
V̄ − rH = 0,
∂H
N (t)

where ξA (t) ≡ KHA . From these two equations, we have
ṽ =

ξAγ
r,
(1 − γ)δ

and rH =

γξA
r.
1−γ

Thus, for the steady state, ξA = (const) is also necessary in this case. Because
H0 is constant, KA also is constant, and we denote it as K̄A . However, the23

growth rate under K̄A is given asgA (t) =K̄Aξγ .
A(t)N (t) A

Therefore, after the long run, the growth rate converges to limt→∞ gA (t) = 0;
namely, the system asymptotically deceases the growth rate and converges
to the Solow regime.

A2. Command Economy and Optimal Conditions
To obtain the welfare properties of the decentralized solution, we consider thesocial planner formulation of this growth model. A benevolent governmentis assumed to maximize the representative household’s utility function Eq.
(8). Therefore, the Hamiltonian of this government can be written as
H(t) =

c(t)1−σ − 1+ λ(t)(η −α KY (t)α N (t)1−α A(t)1−α −c(t)N (t) − IH (t))
+
,.
1−σ
1−γ

+ µ(t)δ

(K(t) − KY (t))
N (t)

Y
γ

H(t)

+ λH (t)IH (t),where λ, µ, and λH are the shadow prices of per capita capital stock, knowledge, and human capital, respectively. The optimal conditions are obtained
as follows:
λ(t)N (t) = c(t)−σ ,(31)

λ(t) = λH (t),'(1−αλ(t)αη −α KY (t)α−1 A(t)N (t)=(32)(K(t) − KY (t))−γ H(t)γ(1 − γ)µ(t)δ
,N (t)(K(t) − KY (t))−γ H(t)γρλ(t) − λ̇(t) = (1 − γ)µ(t)δ
,N (t)(K(t) − KY (t))1−γ H(t)γ−1
ρλH (t) − λ̇H (t) = γµ(t)δ
,N (t)ρµ(t) − µ̇(t) = λ(t)(1 − α)η −α KY (t)α N (t)1−α A(t)−α .

24

(33)
(34)
(35)
(36)

Using Eqs. (31), (34), and (36) and the notation k̃Y , we derive the following
equations:ρ − gλ (t) = αη −α k̃Y (t)α−1 ,1−αρ − gµ (t) =δΓ(γ)k̃Y (t).
α

(37)
(38)

(34) and (35) yield the optimal rate between physical capital KY andγ
human capital H as H = 1−γ
KY ; namely, we have the rate between physicalcapital and human capital, ξ, in both the market and command economies inthis study. Using ξ, the growth rate can also be written as gA = δ(k̃ − k̃Y )ξ γ .
From Eqs. (31) and (37) and the definitions of c̃ and k̃, the following
Euler equation is obtained:)* 4
1 3 −αgc̃ (t) =αη k̃Y (t)α−1 − ρ − n − σδ k̃(t) − k̃Y (t) ξ γ .
(39)σFrom Eqs. (33), (37), and (38), eliminating λ and µ yields
121δ(1 − α)Γ(γ)−αα−1gk̃Y (t) =k̃Y (t) + n − αη k̃Y (t)
.
1−α
α

(40)

The system consists of the following dynamic equations: (2), (39), and(40). These equations imply that gk̃Y = gk̃ and gA = gk = gc = δ(k̃ − k̃Y )must hold in a steady state. Substituting these conditions into Eqs. (2),(39), and (40), we obtain the following equations denoting the steady stateof the command economy:
ρ + σδ(k̃ ∗op − k̃Y∗op )ξ γ = αη −α k̃Y∗op α−1 − n,'(
η −α k̃Y∗op α − c̃∗op − n + δ(k̃ ∗op − k̃Y∗op )ξ γ k̃ ∗op = 0(1 − α)Γ(γ) ∗opαη −α k̃Y∗opα−1 − n =δ k̃Y .
α

(41)
(42)
(43)

By eliminating αη −α k̃Y∗op α−1 −n from (41) and (43), we obtain the result that
optimal capital allocation in the steady state in a command economy is thesame as that in a market economy given by (12). Furthermore, the growthrate for given R&D-difficulty-adjusted capital stock in a command economy
is the same as that in a market economy given by (13).

25Acknowledgement The author thanks Kazuo Mino, Koichi Futagami,
Akihisa Shibata, Katsuya Takii, Akira Momota, Yasutomo Murasawa, Yoichi
Gokan, Katsunori Yamada, and Yui Nakamura for their many useful comments and suggestions. Last, the financial support from the Japan Society
for the Promotion of Science (JSPS Wakate B) is gratefully acknowledged.
Any remaining errors are the sole responsibility of the author.

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Political Economy 103:759-784[17] Jones CI (2002) Introduction to Economic Growth 2nd Edition. W.W.
Norton & Company, Inc.
[18] Kelley AC (1988) Economic consequence of population change in thethird world. Journal of Economic Literature 26:1685-1728[19] Kelley AC, Schmidt RM (1995) Aggregate population and economic
growth correlations: The role of the components of demographic change.
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[21] Kuwahara S (2007) The mechanics of economic growth through capital accumulation and technological Progress. The Japanese Economic
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27

Table 1: Three Types of Steady States
Equilibrium Type
(Condition)A(δ < Ψop )Market Economy
Command Economy

SSS
SSS

op

(Ψ

B
C
< δ < Ψ) (Ψ < δ)
SSS
RSS

RSS
RSS

[23] Matsuyama K (2001) Growing through cycles in an infinitely lived agent
economy. Journal of Economic Theory 100:220-234[24] Quah D (1996) Empirics for economic growth and convergence. European Economic Review 40:1353-1375[25] Quah D (1997) Empirics for growth and distribution: Polarization, stratification, and convergence clubs. Journal of Economic Growth 2:27-59[26] Romer PM (1990) Endogenous technological change. Journal of Political
Economy 98:S71-S102[27] Segerstrom, PS (1998) Endogenous growth without scale effects. American Economic Review 99: 1290-1310[28] Solow RM (1957) Technical change and the aggregate production function. Review of Economics and Statistics 39:312-320[29] Young A (1995) The tyranny of numbers: Confronting the statisticalrealities of the East Asian growth experience. Quarterly Journal of Economics 110:641-680[30] Zilibotti F (1995) A Rostovian model of endogenous growth and underdevelopment traps. European Economic Review 39:1569-1602

28

L(k̃), R(k̃)

L(k̃; α, δ, ρ, n, σ)•#$
Rop (k̃; α, η, δ, ρ, σ) = α1 R(k̃)•
1−αR(k̃; α, η, δ, ρ, σ)

α
−ρ

k̃

k̃ ∗D k̃ ∗op

Figure 1: Equilibrium Stock of the R&D-Difficulty-Adjusted Capital in
Steady Growth

gk̃Y (t)δΓ(γ)
nk̃Y + 1−α
α

gk̃Y
•
k̃Y∗

k̃Y (t)2α− 1−αη −α k̃Yα−1

Figure 2: Dynamics of k̃Y29c̃Solow plane (k̃ = k̃Y )k̃˙ = 0 Linec̃˙ = 0 Linec̃RC

• Steady State
(RSS)

k̃Y∗

Romer plane(k̃Y = k̃Y∗ )

k̃ ∗

k̃A∗

Solow Regimek̃Y∗

Romer Regime45’-Linek̃Y

Figure 3: Global Phase Dynamics (Long-run Growth Case)

30

k̃

c̃
Solow plane (k̃ = k̃Y )c̃˙ = 0 Linek̃˙ = 0 Line
(Romer plane)•

Steady State (SSS)k̃ ∗∗(k̃Y∗ )k̃ ∗∗(k̃Y∗ )
Solow Regime45’-Linek̃Y

Figure 4: Global Phase Dynamics (No-growth Case)31k̃