GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.32

Education, Innovation, and Long-Run Growth

Katsuhiko Hori and Katsunori Yamada

February 2009: revised March 2009; revised November 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Education, Innovation, and Long-Run Growth∗
Katsuhiko Hori†

Katsunori Yamada‡

November, 2011

Abstract
This study augments a second-generation Schumpeterian growth model
to employ human capital explicitly. We clarify the general-equilibrium interactions of subsidy policies to R&D and human capital accumulation ina unified framework. Despite a standard intuition that subsidizing thesegrowth-enhancing activities is always mutually growth promoting, we findasymmetric eﬀects for subsidies on R&D and those on education. Our theoretical result of asymmetric policy eﬀects provides an important empiricalcaveat that empirical researchers may find false negative relationships between education subsidies and the output growth rate, if they merely rely on
the standard human capital model.
JEL classification: O15; O32; O41
Keywords: Schumpeterian growth model; human capital accumulation; subsidies∗

We wish to thank Ronald Harstad, Akiomi Kitagawa, Fumio Ohtake, Akihisa Shibata, John
Stachurski and participants in the annual conference of Western Economic Association International for their helpful comments and suggestions. We would like to acknowledge the research
grant provided by the Global Center of Excellence programs of Keio University and Osaka University.
†
Corresponding author: Institute of Economic Research, Kyoto University. Address: Yoshida
Hon-machi, Sakyo-ku, Kyoto 606-8501, Japan. E-mail address: hori@kier.kyoto-u.ac.jp‡
Institute of Social and Economic Research, Osaka University. Address: Mihogaoka 6-1,
Ibaraki 567-0047, Japan. E-mail address: kyamada@iser.osaka-u.ac.jp1

1

Introduction

A growing consensus in the literature of endogenous growth theory is that Schumpeterian growth theory (e.g. Dinopoulos and Thompson [1998], Peretto [1998],
and Howitt [1999]) is more consistent with empirical facts than earlier versions ofendogenous growth theory including semi-endogenous growth theory.1 Well after the first critique against the semi-endogenous growth theory by Jones [1995],
Ha and Howitt [2007] compare two growth models and show that the data ismore supportive of the Schumpeterian endogenous growth theory than the semiendogenous growth theory. Recent findings further reinforce the finding, showingthat a Schumpeterian prediction that per capita R&D input determines the output growth rate can be seen in the U.S. (Zachariadis [2003]), in OECD countries(Madsen [2008]), and in developing countries (Ang and Madsen [2011]).2
Despite this data-consistent feature of the theory, there is one missing pointin the Schumpeterian endogenous growth theory, which is that usually roles ofhuman capital are not explicitly taken into account in the models. This omission
exhibits a contrast with a traditional view that human capital accumulation is an
important engine of the growth (Uzawa [1965], Lucas [1988], and Rebelo [1991]).
Recently, Madsen [2010] examined an extended version of conventional growthaccounting with data the OECD countries, showing that output growth has been
predominantly driven by total factor productivity (TFP) growth. He then showedthat TFP, in turn, has been driven by R&D and by educational attainment, among
others. His estimates in the growth accounting exercises suggested permanent
growth eﬀects of R&D and human capital.
In order to supplement the gap, this study augments a second-generation Schumpeterian growth model to employ human capital explicitly. The main contributionof the paper is to clarify the general-equilibrium interactions of subsidy policies tothese two engines of growth, R&D and human capital accumulation, in a unified
framework. We see how innovation and education activities interact, and discern
implications for subsidy policies. Despite a standard intuition that subsidizing
growth-enhancing activities is always mutually growth promoting, we show thatsubsidies on the education sector and those on the innovation sector exhibit asymmetric eﬀects. Namely, a subsidy to education has a positive impact on inno1

Semi-endogenous growth theory such as Jones [1995], Kortum [1997], Segerstrom [1998],
and Young [1998] predicts that “the long-run rate of TFP growth, and hence the long-run growthrate of per capita income, depends on the rate of population growth, which ultimately limits thegrowth rate of R&D labor, to the exclusion of all economic determinants.” (p. 734, Ha and Howitt[2007])
2
Madsen, Ang, and Banerjee [2010] also show that innovative activity was an important force
in shaping the Industrial Revolution and that the British growth experience after the Malthusiantrap is consistent with Schumpeterian growth theory.

1vation as well as on human capital accumulation, while a subsidy to innovation
may not aﬀect, or even negatively aﬀect, human capital accumulation.3 Our theoretical result of asymmetric policy eﬀects provides an important empirical caveatthat empirical researchers may find false negative relationships between education
subsidies and the output growth rate, if they merely rely on the standard humancapital model.
There are also theoretical studies considering education and innovation at thesame time. Nelson and Phelps [1966] is the first paper showing that enhancementof human capital accumulation promotes innovation. Redding [1996] and Acemoglu [1997] consider human capital accumulation and R&D in search-theoretic
models with linear preferences and binary decision making, and show that multiplicity of equilibria is a consequence of the complementarities between education and innovation. Arnold [1998] constructs an endogenous growth model withthese two sources of R&D and human capital accumulation. However, the effects of any subsidy policy on economic growth would vanish in the long run,since it is essentially based on a semi-endogenous growth model. Recenly, Zeng
[2003] constructs a similar model to our model and investigate the eﬀects of subsidy policies on output growth, but does not consider the asymmetric eﬀects ofthese policies on engines of economic growth. Connolly and Peretto [2003] also
construct an endogenous growth model close to ours to investigate interactionsof fertility choices and innovation. In their model, endogenous fertility choices
aﬀect per capita output growth rate through the channel of labor inputs. Morerecenly, Grossmann [2007] investigates the eﬀects of public education and R&D
subsidy to innovation, and shows that R&D subsidy may reduce the R&D investment when there are congestion externalities on public education. Along the linesof these studies, we construct a second-generation Schumpeterian growth modelwith human capital accumulation, and clarify the general-equilibrium interactionsof subsidy policies between R&D and human capital accumulation.
This paper is organized as follows. Section 2 presents the baseline model3

As is well known, an important theoretical result of the Schumpeterian endogenous growththeory is that subsidizing R&D activities can have positive long-run eﬀects on the output growthrate, not an implication of semi-endogenous growth theory. Direct empirical evidence supporting
the theoretical results are scarce. Using Japanese and the U.S. data sets, Sakakibara and Branstetter [2001] showed that patent reform policies did not lead to increase in either R&D spending or
innovative output, but they did not work on subsidy policies. Zachariadis [2004] and Ulku [2007]
showed that in OECD countries there are positive relationship between R&D intensity and thegrowth rate of output. However, their conclusions were not from natural experiments of exogenous subsidy policy shocks. Similarly, standard human capital models also allow for policies thatenhance economic growth, but empirical evidence on eﬀects of these policies on the output growthrate is scarce. Instead, the eﬀects of increases in minimum education due to school reforms in European countries over 1960–1980 are well examined. See Machin, Pelkonen, and Salvanes [2008]
and references therein for this topic.

2

where households have logarithmic preferences. Section 3 generalizes the baseline model to allow for general constant intertemporal elasticity of substitution(CIES) felicity function. We show that main results in the baseline case remain.
Section 4 extends the baseline model where only human capital is required forintermediate goods production and vertical innovation needs final goods only. In
the extended model, these activities use both human capital and final goods. It isshown that our main results remain. Finally, Section 5 concludes the paper.

2

The Baseline Model

The production structure of the baseline model is the same as that of Howitt [1999]
except that the raw labor input in the intermediate goods production is replaced byhuman capital in our baseline model. We also employ an Uzawa-Lucas-Rebelotype process for human capital accumulation. Population of the economy is assumed constant over time and normalized to one. The final goods output, whichis the numéraire in the model, is used for consumption (Ct ) and investments forvertical innovation (ZAt ), horizontal innovation (ZNt ), and human capital creation(Zht ).

2.1

Production

The final goods sector operates under perfect competition. The technology forfinal goods production is given by∫ Nt
1−αYt = X
Ait xitα di,(1)0where Yt is the final goods output, X is a fixed resource such as land, Nt measures the varieties of intermediate goods at time t, Ait is a productivity parameterattached to the incumbent version of intermediate product i ∈ [0, Nt ), xit is theamount of intermediate product i used in the economy, and α ∈ (0, 1) is a parameter for production technology. Since we assume that the total endowment of the
fixed resource in the economy is equal to 1, we henceforth abbreviate variable X
from the production function. Under perfect competition, the first-order condition
for the final goods sector with respect to xit is given as
pit = αAit xitα−1 ,(2)where pit is the price of intermediate product i.
The intermediate goods sector operates under monopolistic competition. Here,
we assume that one unit of intermediate good is made from one unit of humancapital.
3Profits Π in creating intermediate product i are given byΠit = pit xit − wt xit ,where wt is the real wage for human capital. With the demand function of xit
obtained from (2), the first-order condition with respect to xit gives the amount of
the ith intermediate good as
1

2

−1

xit = α 1−α Ait1−α wt1−α .

(3)Then, profits of intermediate goods firms can be written as
2α

1

α

Πit = α(1 − α)α 1−α Ait1−α wtα−1 .

2.2

(4)

Innovation

Here, we consider two types of innovation. The first is vertical innovation, which
improves productivity in each intermediate goods sector. The second is horizontal
innovation, which brings new varieties into the economy.
2.2.1

Vertical Innovation

The Poisson arrival rate of vertical innovation in each sector is defined as
φt = λA

ZAt≡ λA zAt ,
At Nt

(5)

where λA > 0 is the productivity parameter of vertical innovation, and ZAt is theamount of resource devoted to vertical innovation for each sector i. At captures
the leading-edge productivity parameter with At ≡ max{Ait | i ∈ [0, Nt ]}. Hence,zAt ≡ ZAt / (At Nt ) is the per-sector expenditures on vertical innovation adjusted byproductivity and complexity. The free-entry condition into vertical innovation is
given byZAt
λA zAt Vt = (1 − sR ),(6)
Ntwhere Vt is the expected present value of a vertical innovation at time t from thestream of future profits. Here, sR ∈ [0, 1) is the general subsidy rate to R&D,
which is an important policy parameter in this study.
Because at every time t, the innovation will be replaced by the next innovatorwith the Poisson arrival rate φt , Vt is given as
[ ∫ τ
]
∫ ∞
Vt =
exp −
(r s + λA zAt )ds πτt dτ,
(7)
t

t

4

where r s is the interest rate, and πτt is the profit of the incumbent on date τ for any
sector with vintage technology at time t, which can be expressed as
2α

1

α

πτt = α(1 − α)α 1−α At1−α wτα−1 .
Further, we can define the quality-adjusted value of vertical innovation as vt ≡Vt /At . Then, the free-entry condition (6) is rewritten asvt =1 − sR.
λA(8)Finally, the intensity of the quality improvement for each vertical innovation
is captured by a parameter σ > 0, with which the growth rate of the leading-edge
productivity is given as
gAt = σλA zAt ,(9)where gAt denotes the growth rate of the leading-edge productivity, gAt = Ȧt /At .
Hereafter, we denote the growth rate of a variable χt by gχt . Therefore, by diﬀerentiating (8) with respect to t and using (7), (8), and (9), a little algebra provides
the expression of the interest rate as([ασα ) ]
2αrt = λA (1 − sR )−1 α(1 − α)α 1−α ωtα−1 − 1 +
zAt ,
(10)
1−α
where ωt denotes the quality-adjusted wage rate (ωt ≡ wt /At ).
2.2.2

Horizontal Innovation

The variety of intermediate goods can be augmented by horizontal innovation.
The evolution of varieties is specified asṄt = λN

β 1−β
Yt
ZNt
,
At

where λN > 0 and β ∈ (0, 1) are parameters of horizontal innovation. ZNt is
the amount of numéraire devoted to horizontal innovation. Knowledge spillover
eﬀects captured by the term Yt is also included. Therefore, the growth rate of thevariety of intermediate goods is given as
gNt = λN

β 1−β
ZNt
Yt
.
At Nt

(11)

Each horizontal innovation results in a new intermediate product whose productivity is randomly drawn from the distribution of existing intermediate products. Further, from the definition of the value of the leading-edge intermediate5

good (At ) given by (7), and from the definition of the profits of intermediate good
firm with quality Ait given by (4), the expected value of a horizontal innovation
is derived as E[(Ait /At )1/(1−α) ]Vt . Hence, the free-entry condition in the horizontal
innovation sector read as
( ) 1  Ait 1−α  Vt = (1 − sR ) ZNt .
Ṅt E 
(12)

At
Then, plugging the definition of Ṅt into the above equation, we get) 1 
β 1−β (ZNt
Yt
 Ait 1−α  Vt = (1 − sR ) ZNt .
λN
E 
At
At

(13)

From the specification of vertical innovation, the distribution of the relative
productivity Ait /At converges to the time-invariant distribution function F(q) =q1/σ , where 0 < q ≤ 1.4 Hence, in the long run, we obtain
( ) 1  Ait 1−α  1
 = ,
E  ΓAtwhere Γ ≡ 1 + σ/(1 − α) > σ. Throughout this paper, we assume that the relative
productivity Ait /At is drawn from the time-invariant distribution function. From(8), (13), and the definition of Γ, we get()1
ZNt
λN −1 1−β
=
Γ
≡ ζ.
Yt
λA

(14)

Note that ζ ∈ (0, 1) must hold because of a resource constraint, Yt > ZNt . To
guarantee that ζ ∈ (0, 1) holds, we impose the following assumption.
Assumption 1. We assume that λA ∈ (λ0A , ∞), whereλ0A =Then, (11) is rewritten as

λN.
Γ

gNt = λN ζ β yt ,(15)

where yt is the amount of output adjusted by the quality and variety (yt ≡ Yt /(At Nt )).

4

See Howitt [1999] and Segerstrom [2000] for the proof.

6

2.3

HouseholdFor the baseline analysis, we assume that the felicity function of household is inlog. The maximization problem of a representative household is given by∫ ∞max
exp[−ρt] log Ct dt0where ρ > 0 is the subjective discount rate, and Ct is the per capita consumption.
The laws of motion for financial assets and human capital in per capita terms arerespectively given asẆt = rt Wt + wt ut ht − Ct − (1 − sh )Zht − T t ,
and

(

Zht
ḣt = λh
At

(µt )

)γ
[(1 − ut )ht ]1−γ ,

(νt )

(16)

where Wt denotes the per capita asset, ht is the per capita amount of human capital,ut ∈ [0, 1] is the ratio of human capital devoted to the intermediate goods sector,
Zht is the expenditure on human capital accumulation, γ ∈ [0, 1) is a parameter forhuman capital accumulation process, and sh ∈ [0, 1) is the other important policy
parameter of the subsidy rate to expenditure on human capital accumulation. Costate variables of µ and ν are attached to the respective constraints. In (16), we
divide the amount of expenditure (Zht ) by At following the idea that the higher
the leading-edge quality in the economy, the more diﬃcult may be the acquisitionof new skills to handle the cutting-edge technology (Howitt [2005]). Finally, theexpenditure of the government is financed by lump sum tax (T t ), and the budgetof government is balanced at any time.
From the above specifications, the first-order conditions of the problem are
obtained as
Ct−1 = µt−γ γµt wt = νt λh (1 − γ)(1 − ut )−γ A−γt ht Zht

(17)
(18)

1−γ γ−1
1−γµt (1 − sh ) = νt λh γA−γt ht Zht (1 − ut )µt rt = ρµt − µ̇t

(19)
(20)

−γ γ
µt wt ut + νt λh (1 − γ)(1 − ut )1−γ A−γt ht Zht = ρνt − ν̇t .

(21)

and

In addition, the usual transversality conditions (TVCs) are imposed on W and h.
7

Combining (17) through (21), the first-order conditions are reduced to the following equations.
gct = rt − ρ − gAt − ght ,(22)

gωt = (1 − γ)−1 (rt − gAt ) − λh γγ (1 − γ)−γ (1 − sh )−γ ωγt ,ght = λh γγ (1 − γ)−γ (1 − sh )−γ (1 − ut )ωγt ,

(23)
(24)

and
zht =

γ(1 − ut )ωt .
(1 − γ)(1 − sh )(25)

Here, zht denotes expenditure to education adjusted by the quality and humancapital (zht = Zht /(At ht )) and gct denotes the growth rate of consumption adjusted
by quality and human capital (ct = Ct /(At ht )). Finally, we define the growth rateof human capital per variety as lt ≡ ht /Nt . Hence,glt = ght − gNt .

2.4

(26)

Equilibrium

The market clearing conditions for final good and human capital market at eachtime t read, respectively, as
Yt = Ct + Zht + ZAt + ZNt ,

(27)

and
∫ Nt

xit di = ut ht .

(28)

0

Finally, from (1), (3), and the definition of yt , we haveαyt = α 1−α Γ−1 ωtα−1 .
2α

(29)

The equilibrium dynamics of the economy is defined as follows.
Definition 1 (Equilibrium Dynamics). An interior equilibrium dynamics is defined as a sequence of prices, (wt , rt , pit ), and allocation, (ZAt , ZNt , Zht , xt , Ct , Yt , ut ),that satisfies (i) profit maximization: (1), (2), (3), (10) and (14), (ii) utility maximization: (22), (24), (25), and the transversality conditions with u ∈ (0, 1),
gA∗ ∈ (0, ∞) (iii) market clearing: (27) and (28), and (iv) dynamics: (9), (15), and(23), given A0 , N0 , h0 , and the distribution of Ai0 /A0 which is given by F(q) = q1/σ ,where 0 < q ≤ 1.
8

2.5

Steady State

2.5.1

The Existence and Uniqueness of the Steady State

Hereafter, we focus on the interior steady state of the equilibrium dynamics, where
gct = gωt = glt = 0 for all t. We add subscript ∗ to any variable whenever it isconstant in the steady state.
To begin with, it follows from (29) and from gω∗ = 0 that the steady-state
growth rate of y is equal to zero. Hence, the definition of y gives the economic
growth rate in the steady state as
gY∗ = gA∗ + gN∗ .

(30)

Since in the steady state it holds that glt = 0, (26) implies that gN∗ = gh∗ . Hence,(30) can be rewritten as the sum of the growth rate of leading-edge quality andthat of human capitalgY∗ = gA∗ + gh∗ .
(31)Equation (31) provides a structural basis of growth regressions ever since the workof Mankiw, Romer, and Weil [1992]: while the arguments in production functions(technology, labor, physical capital, and human capital) are counted in an ad-hoc
manner in the growth empirics literature, our model provides market structuresin which all of these production components are contributing to the economic
growth.
Regarding the growth rate of human capital, firstly we have from (22) thatr∗ = ρ + gA∗ + gh∗ .
This relation is considered the steady-state version of the Euler equation.
From (23), we have()ω∗
r∗ = ξ
+ gA∗ ,
1 − sh(32)(33)

where ξ (χ) = λh γγ (1−γ)1−γ χγ . Note that ξ0 (χ) > 0 for all χ > 0 and limχ→∞ ξ(χ) =∞.
Combining (32) and (33), we have
(
)
(
)
ω∗
ω∗
gh∗ = ξ− ρ ≡ gh.
(34)
1 − sh1 − shSince (34) is derived only from household’s optimal conditions, it can be referred
to as the supply-side growth rate of human capital accumulation.
Secondly, from (29) and (15), we haveα2αgN∗ = λA ζα 1−α ω∗α−1 ≡ gN (ω∗ ) ,9(35)

where g0N (χ) < 0 for all χ > 0. Since the above equation is derived only fromfirm’s optimal conditions, it can be referred to as the demand-side growth rate ofhuman capital accumulation.
Combining (34) and (35) determines the steady-state growth rate of humancapital gh∗ and the quality-adjusted wage rate ω∗ :
(
)
ω∗
(H)
gh
= gN (ω∗ ) .
1 − shSince the left-hand side of the above equation is increasing in ω and the right-handside is decreasing, it determines the unique steady-state value of ω.
Next, we calculate the steady-state values of u and gA∗ . Combining (24) and(34), we haveρu∗ = γ + (1 − γ) ( ω ).
(U)
ξ 1−s∗ h
Substituting (9), (10), and (35) into (32) yields
{ [
]
}
α
α(1 − α)
σ
2α
α−1
1−α
λA
− ζ α ω∗ − ρ .
gA∗ =Γ
1 − sR(A)

We have then the following proposition.
Proposition 1 (Steady State). In the steady state of the equilibrium dynamics, ω∗ ,u∗ , gh∗ , and gA∗ are given by (H), (U), and (A).
Finally, we give the following proposition showing the existence and uniqueness of the steady state.
Proposition 2 (The Existence and Uniqueness of the Steady State). There exist
λA ∈ (0, ∞) and ρ̄ ∈ (0, ∞) such that, for any λA ∈ (λA , ∞) and ρ ∈ (0, ρ̄), there is
the unique steady state of the interior equilibrium dynamics.
Proof. See Appendix A.
In the remaining of this paper, we assume the interior steady state.
2.5.2

Subsidy policies in the baseline model

We provide subsidy policy implications for the baseline model as Theorem 1 andTheorem 2. Despite a standard intuition that subsidizing these growth-enhancing
activities is always mutually growth promoting, we find asymmetric eﬀects for
subsidies on R&D and those on education.

10

Theorem 1. The subsidy to human capital accumulation has positive eﬀects on
the growth rates of the leading-edge technology and human capital.

and

dgA∗
>0
dsh

(36)

dgh∗
> 0.
dsh

(37)

Hence, we have
dgY∗
> 0.
dsh
Proof. Diﬀerentiating both the sides of (H), we obtain( ω )∗g0h 1−sω∗dω∗( ( ω ) h
) < 0.
=−dsh(1 − sh ) g0h 1−s∗ h − (1 − sh ) g0N (ω∗ )Therefore, noting that the left-hand side of (H) is equal to gh and the right-handside is decreasing in ω, we obtain (37). Similarly, since from (A), zA∗ is decreasingin ω, it gives (36). The proof of the last statement in the theorem is straightforward.

)
(
ω∗
g
0
h 1−s
g (ω )
hgh N ∗( ω )∗gh 1−s
h
g0h
gh

O

ω0∗

ω∗

ω

Figure 1: Eﬀects of subsidy to human capital accumulation (s0h > sh )11

Intuitively, the reason for the indirect positive impact of subsidy to human capital accumulation process on the R&D sector is explained as follows. The subsidy
to human capital accumulation raises the growth rate of the supply-side humancapital through (34), which reduces the adjusted wage rate to maintain the equality of (H). When the adjusted wage rate of human capital declines, the adjusted
profit and the expected value of firms (7) increase, motivating the vertical and horizontal innovations. Hence, boosting human capital accumulation augments thegrowth rate of products quality, which in turn accelerates the growth of per capita
output. If we consider only the relationship between the growth rate of output
and R&D investment, as in standard Schumpeterian growth models, we miss thegrowth-enhancing eﬀects of the subsidy policies coming by way of human capital
accumulation process.
Figure 1 depicts the eﬀect of subsidy to human capital accumulation on ω∗ .
In the figure, the gN curve depicts the right-hand side of (H). Two gh curves
respectively plot the left-hand side of (H) corresponding to the subsidy rates ofhuman capital accumulation, sh and s0h . Since gh curve is increasing in sh , ands0h is assumed to be greater than sh , the curve corresponding to s0h lies above that
corresponding to sh . Hence, the figure shows that the increase in the subsidy rate
on human capital raises the growth rate of human capital accumulation gh∗ .
Theorem 2. The subsidy to R&D has a positive eﬀect on the growth rate of theleading-edge technology, but does not have any eﬀect on the growth rate of humancapital.
dgh∗
dgA∗
>0
and
= 0.
dsR
dsR
Therefore, we have
dgY∗
> 0.
dsR
Proof. The diﬀerentiation of (A) with respect to sR gives
α
dgA∗
2α
= σΓλA (1 − sR )−2 α(1 − α)α 1−α ω∗α−1 .
dsR

(38)

Here, it should be noted that ω∗ is determined by (H), which is independent of sR .
Since the right-hand side of (38) is positive, we obtain the theorem. The proof ofthe last statement in the theorem is straightforward.
The reason that subsidy to R&D investment has no eﬀect on human capital
accumulation is as follows. In the steady state, an increase in the subsidy rate
to R&D investment raises both the real interest and productivity growth rates by
the same amount, since we assume the logarithmic preferences. These increases
have two eﬀects on human capital accumulation process. The first is the eﬀect12

induced by the increase in the interest rate, shifting human capital resource toproduction. The other is the eﬀect induced by the increase in the productivity
growth rate, shifting human capital resource to education, since the increase of
productivity growth rate implies the increase of the wage rate in the future. Since
in the baseline case we assume the logarithmic preferences, these two eﬀects oﬀset
each other, leaving u∗ and ω∗ unchanged. Hence, subsidy to R&D investment has
no eﬀect on the growth rate of human capital.
Before closing this section, we add the following proposition.
Proposition 3. The steady state of the equilibrium dynamics is locally saddle
stable.


Proof. See Appendix B.

Not likely to previous studies such as Redding [1996] and Acemoglu [1997],
where multiple steady states result as a consequence of the complementarities
between human capital and R&D, our model has the unique and locally stablesteady growth path which allows for subsidy policies implications along the path.

3

CIES Preferences

In this section, we address CIES preferences instead of log felicity. We showthat the previous results of asymmetric policy eﬀects remain for the generalizedcase. Our theoretical result of asymmetric policy eﬀects provides an important
empirical caveat. We specify that utility function is given as∫ ∞)η ( 1− 1η
exp[−ρt]
Ct − 1 dt,η−10
where η denotes the intertemporal elasticity of substitution. In the steady state,
the Euler equation is modified from (32) tor∗ = ρ +gA∗ + gh∗
.
η

(39)

The above equation shows that, given the interest rate and the growth rate of human capital, a higher degree of intertemporal elasticity of substitution η yields ahigher technological progress.
Combining (33) and (39), we get
)
)
( (
ω∗
− ρ + (η − 1)gA∗ .
(40)
gh∗ = η ξ
1 − sh13

Equation (40) is the modified version of supply-side growth rate of human capital
accumulation. Notice that gh∗ in this case depends on gA∗ , not likely to the case oflog felicity in (34).
Next, from (9), (10), and (39), we obtain the modified version of the growthrate of technology as
[
]
α
ησ/Γ
g
λA
2α
h∗
α−1
gA∗ =
α(1 − α)α 1−α ω∗ − ρ −.
(41)η − (η − 1) σ/Γ 1 − sRη
Substituting (35) into the above equation, we see that
{ [
]
}
α
ησ/Γ
α(1 − α) ζ 1−α2αα−1− α ω∗ − ρ .
gA∗ =λAη − (η − 1) σ/Γ
1 − sRη(A1)

From (24), (35), (40), and (41) we havegh (ω∗ , sh , sR ) = gN∗ (ω∗ )

(H1)

[
σ]
u =1 − (1 − γ) η − (η − 1)
Γ
[
]
α
1−γ
σ λA
2α
α−1
+ ( ω ) ηρ − (η − 1)α(1 − α)α 1−α ω∗ .
Γ 1 − sR
ξ 1−s∗ h

(U1)

(
)
[
ω∗
σ]
gh (ω∗ , sh , sR ) ≡ η − (η − 1) ξΓ
1 − shασ λA2α+ (η − 1)α(1 − α)α 1−α ω∗α−1 − ηρ.
Γ 1 − sR(42)

and u∗ is determined as
∗

Here,

Therefore, we put the following proposition for the interior steady state of theeconomy with CIES preferences.
Proposition 4. In the interior steady state with CIES preferences, ω∗ , u∗ , gh∗ , andgA∗ are given by (A1), (H1), (U1).
Note that the existence of the interior steady state is guaranteed from the continuity around η = 1 under the assumptions in Proposition 2. We then provide thefollowing proposition.55The proof of the proposition and theorems in this section is given in Appendix C.

14

Proposition 5 (Uniqueness of the Steady State). If there is an interior steady state,
it is unique.
We provide subsidy policy implications for this generalized version of theeconomy as Theorem 3 and Theorem 4. As in the log felicity case, we find asymmetric eﬀects for subsidies on R&D and those on education. In this case, however,subsidy to R&D investment can aﬀect human capital accumulation process in addition to R&D sector. The novelty here is that the indirect eﬀect of subsidy to
R&D can slow down human capital accumulation, depending on the magnitudeof η.
Theorem 3 (Eﬀects of Subsidy to Human Capital Accumulation). In the interior
steady state, we have
dgA∗
> 0,
dsh

dgh∗
> 0,
dsh

and

dgY∗
> 0.
dsh

Theorem 4 (Eﬀects of Subsidy to R&D Investment). In the interior steady state,
we have
dgA∗
dgh∗
>0
and
≶0
for η ≶ 1.
dsR
dsR
And
dgY∗
> 0.
dsR
Theorem 3 is a simple replication of Theorem 1. We obtain Theorem 4 sincein this case the two opposite eﬀects of subsidy to R&D investment on the humancapital accumulation process do not oﬀset each other. An increase of subsidy to
R&D investment raises the real interest rate, r∗ , through (10). The increase in r∗
raises the productivity growth rate, gA∗ , by the smaller (larger) amount than theincrease of r∗ through (39) if η is smaller (larger) than one, given the growth rateof human capital, gh∗ fixed. This implies that, given ω∗ , the growth rate of themarginal return to raise human capital investment, which is given by the righthand side of (33), falls behind (goes beyond) the growth rate of the marginal
return to raise savings, which is given by the left-hand side of (33), causing thedecrease (increase) of gh∗ . The figures 2 and 3 indicate the eﬀects of subsidy to
R&D investment in the cases η < 1 and η > 1, respectively.
Our theoretical result of asymmetric policy eﬀects provides an important empirical caveat. For a thought experiment, now suppose that there is an economy
where the government has launched unexpected subsidies both on the education
sector and on the innovation sector. Assume that deep parameters in the economy
are given such that a subsidy to the innovation sector negatively aﬀects human capital accumulation, and assume also that the subsidy to the R&D sector is so great,
15

gh

(
)
ω
gh 1−s
;
s
R
h(
)
ω
0
gh 1−s;sRhgN (ω)ghg0hOω∗ ω0∗

ω

Figure 2: Eﬀects of subsidy to R&D investment in the case η < 1 (s0R > sR )gh gN (ω)

)
(
ω
0
;
s
gh 1−sR( h)ω
gh 1−s
;
s
R
h

g0h
gh
O

ω0∗ ω∗

ω

Figure 3: Eﬀects of subsidy to R&D investment in the case η > 1 (s0R > sR )16compared to that for education sector, that the net eﬀect of these two subsidies isnegative on the growth rate of human capital. On one hand, when researchers focus on conventional Schumpeterian growth models, they will find that the positive
policy shock on the R&D sector promotes the output growth rate as it is expected,
while the eﬀect is under-estimated because of the omission of the channel of human capital accumulation. This is a typical omission variable problem. On theother hand, when researchers focus on conventional human capital models, empirical researchers find false negative relationships between education subsidies
and the output growth rate: after a positive policy shock on the education sector,
the growth rate in human capital is decreased while the growth rate in the output
is increased. Hence, we provide a caveat for future empirical research examining
the relationship between subsidy policies and economic growth.

4

Extension

4.1

Vertical Innovation with Final Good and Human Capital
InputsIn the baseline model, we examined the case where vertical innovation required
final good input only. Following Eicher and Turnovsky [1999], here we extend
the baseline model to incorporate human capital as inputs for vertical innovation.
We show that our previous results do not depend on the assumption that verticalinnovation needs final good input only.
Instead of (5), this section defines the arrival rate of vertical innovation as(
φt = λA

ZAt
At Nt

)θ (

HAt
Nt

)1−θ
.

Here, ZAt and HAt denote the inputs of final good and human capital to develop
the leading-edge technology at time t, respectively. In this case, equations (H) and(A) are respectively replaced by
)
(
α
ω∗
2α
= Λ (ω∗ ) ζα 1−α ω∗α−1 .
(H2)gh
1 − sh
and

{
[
]
}
α
σ
α(1 − α)
2α
α−1
gA∗ =
Λ (ω∗ )− ζ α 1−α ω∗ − ρ ,Γ
1 − sRwhereΛ (ω) = λA( θ )1−θωθ−1 .
1−θ
17

(A2)

(43)

Equation (H2) uniquely determines the steady state values of ω∗ and gh∗ , sinceΛ (ω) is decreasing in ω.
Since (H2) is independent of sR , gh∗ is not aﬀected by the change of sR . Also,
we see that a rise in sh reduces ω∗ through (H2), increasing the growth rate of theleading-edge productivity gA∗ through (A2). Hence, our extension here does notchange previous results in the baseline model.

4.2

Production of the Intermediate Goods with Final Good andHuman Capital Inputs

The other extension from the baseline model is about production of the intermediate goods that now require final good input in addition to human capital inputs.
We specify the production function of the intermediate goods as
( )δ
ZIit
1−δ
xit =
HIit
.
At
Here, ZIit and HIit denote the firm i’s final good and human capital inputs at timet, respectively. In this case, (H) and (A) can be respectively rewritten as()α(1−δ)α(1−δ)
ω∗αδ
2αgh= λζα 1−α δ 1−α (1 − δ) 1−α ω∗ α−1(H3)
1 − shand{ []}α(1−δ)α(1−δ)σα(1 − α)
αδ
2α
α−1
gA∗ =
λ
− ζ α 1−α δ 1−α (1 − δ) 1−α ω∗−ρ .
Γ
1 − sR(A3)

Similarly to the first extension, (H3) uniquely determines ω∗ and is not aﬀectedby the change of the R&D subsidy rate sR . We can see that the increase in sR does
not aﬀect gh∗ , while the increase in sh raises gA∗ . Hence, previous results in thebaseline model remain.

5

Conclusion

In this paper, we augmented Schumpeterian endogenous growth model to discern
general-equilibrium eﬀects of subsidy policies to R&D and human capital accumulation. We addressed a variant of Schumpeterian endogenous growth model after recent empirical findings of Zachariadis [2003], Ha and Howitt [2007]), Madsen [2008]), and Ang and Madsen [2011]. New findings in Madsen [2010] of
permanent growth eﬀects of human capital have provided direct motivations forthis study, since in the Schumpeterian endogenous growth theory usually roles ofhuman capital are not explicitly taken into account in the models.
18

We showed that, despite a standard intuition that subsidizing these growthenhancing activities is always mutually growth promoting, asymmetric eﬀects
for subsidies on R&D and those on education result. Our theoretical result ofasymmetric policy eﬀects provides an important empirical caveat that empiricalresearchers may find false negative relationships between education subsidies andthe output growth rate. Namely, when the indirect negative eﬀect of R&D subsidies on human capital accumulation dominates the direct positive eﬀect of education subsidies on human capital accumulation, empirical researchers may see thatafter a positive policy shock on the education sector, the growth rate in human capital is decreased while the growth rate in the output is increased. We suggest that
empirical researchers should employ a model where technology improvements by
R&D and human capital accumulation are considered in a unified framework inorder not to miss the general equilibrium eﬀects from one sector to the other.

AppendixA

Proof of Proposition 2

We show that the steady growth path uniquely exists. We first give the following
lemma.
Lemma 1. The values of ω∗ , u∗ , and gh∗ given by (H) and (U) are uniquely determined as ω∗ ∈ (0, ∞), u∗ ∈ (0, 1), and gh∗ ∈ (0, ∞).
Proof. gh∗ > 0 is straightforward from (H), since gN (ω) > 0 for all ω > 0.
The existence of ω∗ is also confirmed by (H) as we have that
)
(
)
(
ω
ω
lim gh< lim gN (ω)
andlim gh> lim gN (ω) .
ω→∞
ω→∞
ω→0
ω→0
1 − sh1 − shThe uniqueness of ω∗ follows from the fact that the left-hand side of (H) is strictlyincreasing and that the right-hand side of the same equation is strictly decreasing.
Finally, we prove u∗ ∈ (0, 1). It is obvious from (U) that u∗ > 0. In order toshow that u∗ < 1, here we define ω such that
(
)
ω
gh
= 0,
1 − shor equivalentlyω ≡ (1 − sh ) ξ−1 (ρ) > 0.

19

Since gh is increasing in ω, (H) shows that ω∗ > ω. At the same time, since (U)implies that u∗ is decreasing in ω∗ , we haveρρ
1 = γ + (1 − γ) ( ω ) > γ + (1 − γ) ( ω ) = u∗ .
ξ 1−shξ 1−s∗ hNext, we prove Proposition 2.
Proof of Proposition 2. By Lemma 1, we see that ω∗ , u∗ , and gh∗ uniquely exist
with ω∗ ∈ (0, ∞), u∗ ∈ (0, 1), and gh∗ ∈ (0, ∞).
In order to show the existence of gA∗ in the range of (0, ∞), take λ00A which
satisfies the condition thatα(1 − α)= ζ,
1 − sR
from which we can explicitly obtain λ00A as[
λ00A =

1 − sRα(1 − α)

]1−β
λN Γ.

[
]
Let λA be max λ0A , λ00A and take any λA ∈ (λA , ∞). Then, for a suﬃciently smallvalue of ρ > 0, we obtain gA∗ > 0 from (A). Define that ρ̄ = supgA∗ >0 ρ. Then,since ω∗ is increasing in ρ from (34) and (H), (A) shows that gA∗ takes a positive
value for any ρ ∈ (0, ρ̄), which completes the proof.


B

Local Stability of the Steady State

Local saddle-stability of the steady growth path is proved as follows.
From (3) and the definitions of ωt and lt , (28) is rewritten as−1

ut = α 1−α Γ−1 ωt1−α lt−1 .
2(44)

From (14), (27), and (29), we get−αzAt = (1 − ζ)α 1−α Γ−1 ωtα−1 − (ct + zht )lt .
2α

(45)

Given the value of ω∗ and u∗ determined by (H) and by (U), (44) gives the steadystate value of l∗ as
12(46)l∗ = α 1−α Γω∗α−1 u−1∗ .

20

With the steady-state values of ω∗ , u∗ , and l∗ , substituting (25) into (45) we haveαc∗ = (1 − ζ)α 1−α Γω∗α−1 l∗−1 +2αγ(1 − u∗ )ω∗ − zA∗ l∗−1 .
(1 − γ)(1 − sh )(47)

The system of equations for the equilibrium dynamics of the model consistingof (22), (23), and (26). Together with supplemental conditions of (9), (10), (15),(24), (25), (29), (44), and (45), we linearize the system around the steady state as ċ  c  c − c∗  ω̇  ω  ≈ J ω − ω∗  ,l˙l − l∗lwhere J is the Jacobi matrix of
 ∂r
∂r
h
A
 ∂c − σλA ∂z∂cA− σλA ∂z
− ∂g
∂ω
∂ω
∂ω
 ∂r∂r
A
J =  ∂c − σλA ∂z∂cA ∂ω− σλA ∂z
− γ(1 − γ)Ψωγ−1
∗
∂ω

∂gh
β ∂y
0
− λN ζ ∂ω∂ω∂r− σλA ∂z∂lA − ∂g∂lh ∂l∂r− σλA ∂z∂lA  ,
∂l

∂gh
∂l

with Ψ ≡ λh γγ (1 − γ)−γ (1 − sh )−γ > 0. After elementary row operations, J can be
reduced to
 ∂r

∂r
∂r
h
A
 ∂c − σλA ∂z∂cA− σλA ∂z− ∂g
− σλA ∂z∂lA − ∂g∂lh 
∂ω
∂ω
∂ω
∂l


∂gh
∂gh
J 0 = 0− γ(1 − γ)Ψωγ−1
 .
∗
∂ω
∂l


γ−1
β ∂y
0
γ(1 − γ)Ψω∗ − λN ζ ∂ω0In order to show the local saddle-stability of the equilibrium dynamics, it issuﬃcient to prove that the determinant of J 0 is negative and the trace is positive.
The determinant of det J 0 is obtained as
(
)[
]
∂r
∂zA
0
γ−1
β ∂y ∂gh
det J = −− λA σγ(1 − γ)Ψω∗ − λN ζ
,
∂c
∂c
∂ω ∂l
where

and

(
ασ )
∂r
= λA 1 + σ +l∗ > 0∂c
1−α∂zA= −l∗ < 0
∂c
1
∂y
α
2α
=
α 1−α Γω∗α−1 < 0,∂ω α − 1

∂gh
2
γ− 1
= α 1−α ΨΓω∗ 1−α l∗−2 > 0.
∂l
21The signs of partial derivatives in the above expressions hold that the determinantis negative.
The trace of J 0 , in turn, can be obtained as
traceJ 0 =∂r∂zA ∂gh− λA σ+− γ(1 − γ)Ψωγ−1
,
∗
∂c
∂c
∂ω

where
∂gh
1
2
γ− 1 −1− γ(1 − γ)Ψωγ−1= γΨω∗γ−1 − (γ −)α 1−α ΨΓω∗ 1−α l∗−1 − γ(1 − γ)Ψωγ−1
∗
∗
∂ω
1−α
1
2
γ− 1 −1= γ2 Ψω∗γ−1 − (γ −)α 1−α ΨΓω∗ 1−α l∗−1 .
1−α
It is easy to show that this expression is always positive, since α ∈ (0, 1) and sinceγ ∈ (0, 1). Hence, the trace of J 0 is positive. These results show that the steadystate of the equilibrium dynamics is locally saddle stable.
C

Proof of Proposition and Theorems in the Case of
CIES Preferences

To begin with, using (35) and (42), we introduce an excess supply function forhuman capital accumulation as
E (ω, sh , sR ) ≡ gh (ω, sh , sR ) − gN (ω)
(
) [
]
[
ω
σ]
σ α(1 − α)α2α= η − (η − 1) ξ
+ (η − 1)− ζ λA α 1−α ω α−1 − ηρ. (48)Γ
1 − shΓ 1 − sR
In the steady state, it should be satisfied thatE (ω∗ , sh , sR ) = 0.

(49)

We then have the following lemma.
Lemma 2. If η ∈ (0, ζ (1 − sR ) / [α(1 − α)]), we havelim E (ω, sh , sR ) = −∞

ω→0

and

and

∂E
(ω, sh , sR ) > 0∂ωlim E (ω, sh , sR ) = +∞,ω→∞for all22ω ∈ (0, ∞).

Proof. Since it holds that ζ (1 − sR ) / [α(1 − α)] < (1 − sR ) ζΓ/ [σα(1 − α)] + 1
and that η < ζ (1 − sR ) / [α(1 − α)], the second term of (48) diverges to −∞ asω → 0 and converges to 0 as ω → ∞. Therefore, since the first term of (48) is
positive, (48) converges to 0 as ω → 0 and diverges to +∞ as ω → ∞.
The last argument is true since
)
(
[
σ] 0
∂E
1
ω
(ω, sh , sR ) = η − (η − 1) ξ∂ω
Γ
1 − sh 1 − sh[]
ζ (1 − sR )
σ λA
2
1
+
− (η − 1)α 1−α ω α−1 > 0.
α(1 − α)Γ 1 − sRProof of Proposition 5. It is straightforward from Lemma 2.

Proof of Theorem 3. By applying the implicit function theorem to (49), we have∂E(ω∗ , sh , sR )
dω∗
∂sh
,
= − ∂Edsh(ω∗ , sh , sR )
∂ω

where

(50)

(
)
[
∂E
ω∗
σ ] 0 ω∗(ω∗ , sh , sR ) = η − (η − 1) ξ
> 0.
∂sh
Γ
1 − sh (1 − sh )2

Notice that (50) is positive by Lemma 2. Therefore, it follows from (H1) thatdgh∗dω∗
= g0N (ω∗ )
> 0.
dsh
dsh
Diﬀerentiating (A1) with respect to sh , we have[]ησ/Γα(1 − α) ζ
α
dgA∗
2α
1 dω∗
=−
− λA α 1−α
ω α−1
> 0.
dsh
η − (η − 1)σ/Γ 1 − sR
η
1−α
dsh

(51)

This holds, since gA∗ > 0 by assumptions which requires a parametric restriction
from (A1) thatα(1 − α) ζ< .
1 − sRη
We naturally obtain dgY∗ /dsR > 0.



Proof of Theorem 4. From (49), we have∂E(ω∗ , sh , sR )dω∗∂s= − ∂ER
,
dsR
(ω
)
,
s
,
s
∗
h
R
∂ω

23

(52)

whereα∂Eσ α(1 − α)2α(ω∗ , sh , sR ) = (η − 1)
λA α 1−α ω∗α−1 .
2∂sRΓ (1 − sR )We show the case of η < 1 because of the space constraint. It is easy to verify
the case of η > 1. Since the numerator of (50) is negative and the denominator ispositive by Lemma 2, we have dω∗ /dsR > 0. Hence, from (H1) we havedgh∗dω∗
= g0N (ω∗ )
< 0.
dsR
dsR

(53)

We also obtain from (40) that
( ω )
dgh∗
1 dω∗
∗
− ηξ0 1−sdgA∗dsR
1−sh dsR
h
=
.
dsR
η−1

(54)

Since dgh∗ /dsR < 0 and dω∗ /dsR > 0, the above equation is found to be positive.
Finally, with dω∗ /dsR > 0, from (31), (53), and (54), we have
(
(
)
)
ω∗
1
dω∗
dgY∗
η
0
0
=
gN (ω∗ ) − ξ
> 0.
dsR
η−1
1 − sR 1 − sR dsR

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