Discussion Papers In Economics
And Business

Basic and Applied Research: A Welfare Analysis
Kunihiko KonishiDiscussion Paper 15-08

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANBasic and Applied Research: A Welfare Analysis
Kunihiko KonishiDiscussion Paper 15-08

May 2015

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANBasic and Applied Research: A Welfare Analysis∗
Kunihiko Konishi†, ‡Abstract
This study constructs a variety expansion growth model that integrates basic research
to analytically examine its eﬀects on household welfare. In our approach, the researchsector consists of applied and basic research components. The former creates blueprints andexpands the variety of goods available for consumption, whereas the latter adds to the stockof public knowledge. The two sectors interplay through knowledge spillovers. The analysis
reveals two key results. First, the steady-state welfare-maximizing level of basic research is
below the steady-state growth-maximizing level. Second, a reduction in the level of basic
research raises household welfare if the level of basic research is initially at the steady-state
welfare-maximizing level.

Keywords: Basic research, Innovation, Endogenous growth, Welfare analysis
JEL classiﬁcation: H41, O31, O41∗

I am especially grateful to Koichi Futagami for helpful discussions and suggestions. I also thank Shiro
Kuwahara, Asuka Oura, and the participants at the 9th Conference of Macroeconomics for Young Professionals
in Osaka for their useful comments. The ﬁnancial support from the Research Fellowship for Young Scientists of theJapan Society for the Promotion of Science (JSPS) is gratefully acknowledged. Any errors are my responsibility.
†
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, JAPAN;
E-mail: nge008kk@student.econ.osaka-u.ac.jp‡
Research Fellow of the Japan Society for the Promotion of Science (JSPS).

1

1

Introduction

Basic research plays an essential role in the innovation process by discovering new knowledge
that is not immediately ready to be commercialized. Applied research then uses the results ofbasic research to develop marketable new products or technologies, such as X-ray, penicillin,
nuclear ﬁssion, packet-switching theory, and methods for DNA sequencing and RNA interference.1
The economic contribution of basic research has been examined in many empirical studies.
For instance, Griliches (1986), examining United States (U.S.) manufacturing ﬁrms in the 1970s,
ﬁnds that basic research has a more signiﬁcant productivity eﬀect than applied R&D. Jaﬀe
(1989) ﬁnds a signiﬁcant eﬀect of university research on corporate patent activity using statelevel time-series data on corporate patents, corporate R&D, and university research. Mansﬁeld(1998) ﬁnds that 15% of new products and 11% of new processes in the U.S. between 1986 and1994 could not have been developed or would have developed with great delay without academic
research. Mansﬁeld (1998) further ﬁnds that the shares of new products and processes that
beneﬁted greatly from academic research are 8% and 7%, respectively. Cohen et al. (2002)
identify a set of very important channels that enable public research to have a positive eﬀect
on industrial R&D: publication, reports, informal information exchange, public meetings orconferences, and consultancy.
The present study examines theoretical policy implications of basic research, particularly on
the welfare of households. To do this, we incorporate basic research into a variety expansion
model, following the work of Grossman and Helpman (1991). In our model, the research sector
consists of applied and basic research streams. The former creates blueprints and develops thevarieties of available consumption goods, while the latter expands the public-knowledge stock.
The productivity of each research activity depends on the existing knowledge that has beenpreviously produced through applied and basic research. We also assume that basic research is
publicly funded—and thus that the government can control the level of basic research. According
to Table 2 in Gersbach et al. (2013), which summarizes 2009 data from a selection of 15
countries, the average share of basic research that was ﬁnanced by governments and higher
educational institutions was 77.39%; that is, basic research is mainly funded by the governmentand carried out at universities or other public research institutions. On the other hand, on

1

See Table 3 in Gersbach et al. (2009) for further examples.

2

average 76.62% of applied research was ﬁnanced by business enterprises and private non-proﬁt
institutions; that is, applied research is primarily performed by private ﬁrms motivated by theirown beneﬁts.
The present analysis obtains two main results. First, the steady-state welfare-maximizing
level of basic research is lower than the steady-state growth-maximizing level.2 The steady-state
growth rate follows an inverted-U shape relationship with respect to the level of basic research.
However, a higher level of basic research increases wages for skilled labor; as a result, goods
prices increase, reducing household consumption. When the government increases the level ofbasic research to maximize the growth rate, this harms household consumption, ensuring thatthe steady-state welfare-maximizing level of basic research lies below the steady-state growthmaximizing level.
Second, reducing the level of basic research increases household welfare if the economy isinitially in the steady state with basic research set at the steady-state welfare-maximizing level.
Although our model exhibits transitional dynamics, we can undertake a welfare analysis using
Judd’s (1982, 1985) method. In our model, an increase in basic research aﬀects welfare throughtwo channels: reducing household consumption and enhancing long-run growth. First, when thegovernment increases the level of basic research, the wage rate initially jumps up and thereafter
monotonically increases to the new steady-state level. This raises the prices of goods, decreasing
household consumption. Turning to the second channel, the eﬀects of increasing basic researchon the growth rate diﬀer between the short and long terms. The short-run growth eﬀect isambiguous, whereas the long-run growth eﬀect is positive. Our analysis shows that the negative
welfare eﬀect can outweigh the positive one, leading to the earlier-stated result.
The theoretical implications of basic research policy have been considered from various
macroeconomic perspectives. The present study is closely related to models in which the basic
research sector is seen as adding to the stock of public knowledge, which in turn raises the
productivity of applied research. Park (1998) considers the growth eﬀect through the interplay
between basic and private research in closed and open economies. In a closed economy, basic research has a positive eﬀect on growth, but increasing basic research also crowds out labor input
into private R&D. In an open economy, the growth rate increases due to international knowledge spillovers, reducing the growth-maximizing level of basic research within a given country.
2
The model exhibits transitional dynamics because there are two state variables: the stocks of knowledge
produced by applied and basic research. We refer to the welfare level at which the economy is constantly in thesteady state as steady-state welfare.

3

Gersbach et al. (2009) assume that basic research generates ideas whereas applied researchcommercializes them by transforming them into blueprints for new varieties of goods—that is,there is a one-to-one relationship between ideas and potential blueprints.3 In this set-up, there
exist two possible equilibria. One is the case in which the growth rate is bound by investments inbasic research. In the other, the growth rate is determined by both basic and applied research.
The prior studies mainly focus on the eﬀects of basic research on the long-run growth rate but
do not examine another key topic: its welfare eﬀects. In contrast, the present study examines
the welfare eﬀects of basic research, taking into consideration transitional dynamics.4
This study is also related to the strand of literature on R&D-based growth models in which
innovation is characterized as a two-stage research activity, with basic research followed byapplied research. In Chu and Furukawa (2013), basic and applied research follow a variety
expansion framework. In Cozzi and Galli (2009, 2013, 2014), however, the two types of researchfollow a quality ladder framework. In Chu et al. (2012), basic research is associated withhorizontal innovation whereas applied research is associated with vertical innovation. In thesemodels, the monopoly proﬁt is divided between those undertaking basic and applied research.
Thus, basic research is motivated by private incentives and thus not funded by the government.
These models are more concerned with the patentability of basic research (i.e., proﬁt divisionrules) than its level.5,6
The rest of this paper is organized as follows. Section 2 establishes the model used in thisstudy. Section 3 derives the equilibrium dynamics of the economy and proves the uniqueness ofthe transitional dynamics. Section 4 analyzes how the policy aﬀects the long-run growth rate
and the transitional dynamics. Section 5 examines the policy eﬀect on steady-state welfare aswell as the welfare eﬀect of marginal changes in basic research. Finally, Section 6 concludes thepaper.

3This setting is similar to those of the below-mentioned models (Cozzi and Galli, 2009, 2013, 2014; Chu etal., 2012; Chu and Furukawa, 2013). However, the reduced form is analogous to Park’s (1998) model.
4
The more simpliﬁed models of Arnold (1997) and Konishi (2013) assume that an increase in the number ofpublic researchers immediately raises the productivity of applied research. In reality, however, it takes time toaﬀect applied research when the government increases investment in basic research. These studies do not consider
the short-run eﬀects of changes in the basic research policy.
5
These studies consider the fact that the U.S. and the European Union gradually extend the patentability ofbasic research. For example, the Bayh-Dole Act, enacted by the U.S. Congress in 1980, permits a university, small
business, or non-proﬁt institution to elect to pursue ownership of an invention in preference to the government.
The European Research Council (ERC) was launched in 2007 to support and promote fundamental researchthrough ensuring the patentability of basic research.
6
See also Gersbach et al. (2013) and Gersbach and Schneider (2015) on the interaction between investment
in basic research and open economy issues.

4

2

Model

The model assumes a unit continuum of identical households, each of which inelastically supplies
one unit of skilled labor and L units of unskilled labor. The factor market is perfectly competitive, and the goods market is monopolistically competitive, as explained below. Households
have perfect foresight.

2.1

Households

Households maximize the following lifetime utility:
Ut ≡

∫ ∞

e−ρ(τ −t) log Cτ dτ,(1)twhere Ct represents instantaneous utility derived from consumption of a composite good andρ > 0 is the time preference rate. Ct is given by
] ε
[∫ Nt
ε−1
ε−1
ε
,
Ct =
ct (j) dj(2)0

where ct (j) denotes the consumption of good j and Nt denotes the number of available varieties.
ε is the elasticity of substitution between any two products, and we assume that ε > 1. Denoting∫N
the consumption expenditure of households as Et = 0 t pt (j)ct (j)dj, we obtain the demand
function for good j as follows:pt (j)−ε Et

ct (j) = ∫ Nt

pt (i)1−ε di

0

,

(3)

where pt (j) is the price of good j and PD,t is the price index, deﬁned as(∫ Nt
PD,t =1−εpt (i)

) 1
1−ε
di
.

0

Substituting (3) into (2), we obtain the indirect sub-utility function as follows:
Ct =

Et
.
PD,t

(4)

Maximizing subject to the intertemporal budget constraint yields the following Euler equation:Ėt= rt − ρ,Et(5)where rt represents the rate of return on assets. Following Grossman and Helpman (1991), we
normalize household consumption expenditures at unity, and thus, Et = 1. As a result, we
obtain rt = ρ.
5

2.2

Firms

Turning to producer behavior, we assume that each diﬀerentiated good that has been createdby applied research is produced by a single ﬁrm because the good is inﬁnitely protected by apatent. We further assume that the production function of good j is a Cobb-Douglas form as
follows:

[
]θ [
]1−θ
xt (j) = ax lts (j) ltu (j),ax > 0 and θ ∈ (0, 1),

where xt (j) is the output of good j, ax is the productivity of production, θ is the intensity ofskilled labor in production, and lts (j) and ltu (j) respectively denote the amount of skilled andunskilled labor devoted to producing good j. With cost minimization, the unit cost functionz(wt , wtu ) is( )θ ( u )1−θ−θθ−1z(wt , wtu ) = a−1
wt wt,
x θ (1 − θ)(6)where wt and wtu represent the wage rates for skilled and unskilled labor, respectively. Applying
Shephard’s lemma, we obtain demand functions for skilled and unskilled labor as follows:θz(wt , wtu )xt (j),
wt(1 − θ)z(wt , wtu )xt (j).
ltu (j) =wtu
lts (j) =(7)(8)

The ﬁrm manufacturing good j (ﬁrm j) maximizes its proﬁt: πt (j) = pt (j)xt (j)−z(wt , wtu )xt (j).
Then ﬁrm j charges the following price:pt (j) = pt =εz(wt , wtu ).
ε−1

(9)

Therefore, all goods are priced equally. Pricing rules (9) and (3) yieldxt (j) = xt =ε−1
1.
ε z(wt , wtu )Nt(10)

Then, the brand-speciﬁc operating proﬁts are given byπt (j) = πt =

2.3

1
.
εNt

(11)

Basic and applied research

Next, we consider the technology involved in developing a new good. Following Park (1998), we
assume that the research sector consists of applied and basic research segments, with appliedresearch creating blueprints and expanding the variety of goods available for consumption and6basic research adding to the stock of public knowledge, Bt . Each research activity requires
skilled labor input. We assume the following applied and basic research technologies:Ṅt = aN Ntα Bt1−α LR,t ,0 < α < 1,

(12)

Ḃt = aB Ntβ Bt1−β Gt ,0 < β < 1,(13)

where aN , aB , LR,t , and Gt represent the productivity of applied research, the productivity ofbasic research, the amount of skilled labor devoted to applied research, and the amount of skilledlabor devoted to basic research, respectively. Each research activity’s productivity depends on
existing knowledge that has been produced through prior applied and basic research. For
simplicity, the knowledge spillover function is assumed to follow a Cobb-Douglas form.7
Basic research is ﬁnanced by a lump-sum tax Tt on households because we do not want tohave to consider the distortionary eﬀects of taxes. That is, the government budget constraint
becomes Tt = wt Gt .8 We assume that applied research ﬁrms freely enter into the R&D race.
The instantaneous proﬁt of these ﬁrms is given by vt Ṅt − wt LR,t , where vt denotes the patentvalue. Consequently, the free entry condition yields
vt =wt⇔ Ṅt > 0.
aN Ntα Bt1−α(14)The shareholders of these ﬁrms’ equities earn dividends and capital gains or losses. Hence, thereturn on equity is given by
rt = ρ =3

Equilibrium

3.1

Dynamic systemπt v̇t
+ .
vt
vt

(15)

For simplicity, we assume that the government keeps the number of public researchers constant(i.e., Gt = G). Skilled labor is used for production, applied research, and basic research. The
market-clearing condition for skilled labor becomesNt lts + LR,t + G = 1.
7

(16)

As noted earlier, Gersbach et al. (2009) assumes a one-to-one relationship between ideas and potential
blueprints. In this set-up, the applied research sector is distinguished in two cases. When Nt < Bt , the production
function for applied research coincides with (12). When Nt = Bt , Ṅt = Ḃt holds and Nt cannot exceed Bt . If
we impose the condition that restricts Nt < Bt , our main results do not change.
8This study focuses on the eﬀects of basic research spending in terms of the number of public researchers.
According to National Science Foundation (2011) analysis of U.S. R&D spending, 46.7% goes to wages for R&D
personnel, 10.1% to employer-sponsored beneﬁts for R&D personnel, 11.7% to materials and supplies, 3.9% todepreciation, and 27.6% to other costs. Therefore, the majority of R&D spending is connected to personnel.

7

The market-clearing condition for unskilled labor isNt ltu = L.
Using (8) and (10), this condition becomes
wtu = wu =(1 − θ)(ε − 1)
.
εL

(17)

t
Therefore, the wage rate for unskilled labor becomes constant. Let us deﬁne λt ≡ N
Bt . In1
addition, (7) and (10) yield Nt lts = θ ε−1
ε wt . (11), (12), (13), (14), (15), and (16) yield

(
)
ε−1 1
ẇt
aN λα−1α−1t
1−G−θ= αaN λt+(1 − α)aB Gλβt −
+ ρ.
wt
ε wt
ε wt
By using (12), (13), and (16), we obtain
(
)
λ̇t
ε−1 1
= aN λα−11−G−
θ−aB Gλβt .
t
λt
ε wt

(18)

(19)

Equations (18) and (19) thus form an autonomous dynamic system with respect to wt and λt .

3.2

Steady state and stability

The steady state is deﬁned by the condition wherein wt , λt , and the growth rates of Nt and Bt
are constant. The equation for ẇt = 0 can be represented bywt =aN {1 + α(ε − 1)θ}εαaN (1 − G) + ε(1 − α)aB Gλt1−α+β + ερλ1−α
t

,

(20)

while the equation for λ̇t = 0 can be represented bywt =(ε − 1)θ
1.
aB
ε1 − G − Gλt1−α+β

(21)

aN

By eliminating wt from equations (20) and (21), the equation that determines the steady-statevalue is as follows:{}Γ(λ∗ ) ≡ 1 + (ε − 1)θ aB G(λ∗ )1−α+β + (ε − 1)θρ(λ∗ )1−α − aN (1 − G) = 0.

(22)

Asterisks represent variables in the steady state. To investigate whether there is a value of λ∗that satisﬁes Γ(λ∗ ) = 0, we diﬀerentiate Γ(λ∗ ) with respect to λ∗ to yield{}Γ′ (λ∗ ) = (1 − α + β) 1 + (ε − 1)θ aB G(λ∗ )−α+β + (1 − α)(ε − 1)θρ(λ∗ )−α > 0.

8

By using Γ(0) = −aN (1 − G) < 0, it is easy to conﬁrm that Γ(λ∗ ) < 0 when λ∗ is suﬃciently
small and Γ(λ∗ ) > 0 when λ∗ is suﬃciently large. There is thus a unique positive value of λ∗that satisﬁes Γ(λ∗ ) = 0. The steady-state value w∗ is obtained from (21) as follows:w∗ =(ε − 1)θ
1.
B
ε1 − G − aaN
G(λ∗ )1−α+βFrom (12) and (16), the growth rate under the steady state is given by()( )∗ ( )∗
Ḃt
ε−1 1
Ṅt
=
= aN (λ∗ )α−1 1 − G − θ
.
γ∗ ≡
Nt
Bt
ε w∗

(23)

(24)

We conﬁrm that w∗ and γ ∗ are surely positive. Using (22), we can rewrite (23) and (24) as
follows:w∗ =1 + (ε − 1)θ
1> 0,
ε1 − G + aρN (λ∗ )1−α
∗

(25)

∗ β

γ = aB G(λ ) > 0.
Next, we examine the stability of the steady state. First, we show in Appendix A that thesteady state is a locally stable saddle point. Second, we investigate the phase diagram of the
dynamic system. Figure 1, depicted in (λt , wt ) space by using (20) and (21), shows that the
only equilibrium is the saddle path that approaches the stable steady state. At the end of thissubsection, we consider the region in which applied research is conducted (i.e., Ṅt > 0). From(16), we obtainṄt > 0 if
Ṅt = 0 if

(ε − 1)θ
,ε(1 − G)(ε − 1)θ
wt ≤.
ε(1 − G)wt >(ε−1)θ
Note that the line wt = ε(1−G)
lies above the line λ̇ = 0. As shown in Figure 1, a suﬃciently

large λt implies that the economy is in the region in which Ṅt = 0. By using (7), (10), (16),and Ṅt = 0, we obtainwt =(ε − 1)θ
.
ε(1 − G)

Hence, if λt is suﬃciently large, wt is constant and λ̇t < 0 holds. Because Bt is relatively small
compared with Nt , there is no incentive to conduct applied research and the remainder of theskilled labor force, 1 − G, is allocated to production. When λt decreases suﬃciently, appliedresearch is conducted and the economy converges to the steady state.99The case where G = 0 corresponds to Grossman and Helpman (1991), Subsection 3.1. If the initial numberof varieties, N0 , is suﬃciently small, the economy is stable at the saddle point. In the transitional dynamics,
applied research is conducted (Ṅt > 0). However, in the long run, growth in the number of diﬀerentiated goods
stops (Ṅt = 0). On the other hand, if the initial number of varieties, N0 , is suﬃciently large, the economy initiallyjumps to the steady state. In this case, applied research is never conducted.

9Figure 1: Phase diagram of the dynamics of wt and λt .

4Eﬀects of policy changes

4.1

Long-run eﬀects

We examine eﬀects of changes in G on the steady-state values. Taking the total diﬀerentials of(22) yieldsaN (λ∗ )α + {1 + (ε − 1)θ}aB (λ∗ )1+β
dλ∗
=−
< 0.
dG
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ

(26)

Thus, λ∗ is decreasing in G. When the government raises G, public knowledge accumulates at
an accelerated rate. As a result, the steady-state ratio of private to public knowledge, λ∗ , falls.
By using (26), we diﬀerentiate (23) with respect to G as follows:[]dw∗ε(w∗ )2 (1 − α + β)aN aB G(λ∗ )β + (1 − α)ρaN + (1 − α)ρaB (λ∗ )1−α+β
=
> 0.
dG
aN
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ

(27)

w∗ is increasing in G. To identify the eﬀects of G on w∗ , we diﬀerentiate w∗ with respect to Gin (25) as follows:{}dw∗ε(w∗ )2ρdλ∗=
1 − (1 − α) (λ∗ )−α.
dG
1 + (ε − 1)θ
aN
dG
There are two eﬀects that increase w∗ . First, an increase in G decreases the amount of skilledlabor devoted to production and applied research. This raises skilled labor demand, and, as
10

a result, the wage rate for skilled labor also increases. Second, a rise in G promotes theaccumulation of public knowledge, and the productivity of applied research increases. This
raises the demand for skilled labor for applied research as well as the wage rate for skilled labor.
We then investigate the eﬀects of G on the steady-state growth rate γ ∗ . Using (26) and(27), we diﬀerentiate (24) with respect to G as follows:(1 − α)(1 − G) − βGdγ ∗= aN aB (λ∗ )α+β−1.
dG(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ

(28)

Equation (28) implies that1−αdγ ∗R 0 ⇔ G Q Gg ≡
.
dG
1−α+β

(29)

Analogous to Park (1998) and Gersbach et al. (2009), the relationship between the steadystate growth rate and G follows an inverted-U shape and there exists a steady-state growthmaximizing level of G. We now study in detail the relationship between the growth rate and G
in the steady state. From (12), the steady-state growth rate, γ ∗ = aN (λ∗ )α−1 L∗R , is determinedby λ∗ and L∗R . Diﬀerentiating γ ∗ with respect to G yields
dL∗
dγ ∗
dλ∗
= −(1 − α)aN (λ∗ )α−2 L∗R+ aN (λ∗ )α−1 R .
dG
dG
dG

(30)

The ﬁrst term represents the growth-enhancing eﬀect and is positive: an increase in G enhances
the accumulation of public knowledge, increasing applied research productivity and thereby thegrowth rate. The second term represents the eﬀect of labor input into applied research; thesign ondL∗R
dG

is ambiguous. From the skilled labor market-clearing condition, labor input into1∗
applied research in the steady state is given by L∗R = 1 − G − θ ε−1
ε w∗ . Diﬀerentiating LR withrespect to G yieldsdL∗R
ε − 1 1 dw∗= −1 + θ.
dG
ε (w∗ )2 dG(31)

The ﬁrst term represents a crowding-out eﬀect on labor input into applied research. The second
term is positive. From (9), an increase in w∗ raises good prices, reducing their demand andreallocating skilled labor from production of goods to applied research. Thus, there is a trade-oﬀ
between these two eﬀects. As shown in Appendix B, we obtain the following relation:dL∗R
R 0 ⇔ G Q G̃,
dG

11

dL∗R
dG G=G̃ = 0. When 0 < G < G̃, the production labor demand eﬀectdL∗
exceeds the crowding-out eﬀect and dGR > 0 holds. In contrast, when G̃ < G < 1, the crowdingdL∗
out eﬀect is suﬃciently large and dGR < 0 holds.10

where G̃ is deﬁned as

Next, we compare Gg with G̃. Substituting G = G̃ into (30) yieldsdγ ∗dλ∗= −(1 − α)aN (λ∗ )α−2 L∗R> 0.
dG G=G̃dG G=G̃
Therefore, from (29), we have G̃ < Gg . These results can be summarized as follows. If 0 <G < G̃, basic research complements applied research, and an increase in G thus increases thegrowth rate. If G̃ < G < Gg , basic research is a substitute for applied research. However, thegrowth-enhancing eﬀect exceeds the crowding-out eﬀect, and an increase in G raises the growthrate. If Gg < G < 1, in contrast, the crowding-out eﬀect exceeds the growth-enhancing eﬀect,
so an increase in G reduces the growth rate.

4.2

Short-run eﬀects

In this subsection, we investigate the transitional dynamics after changes in G. Suppose that theeconomy is initially in the steady state. The phase diagram can then be used to understand the
dynamic paths after an increase in G at time 0. When G rises, the locus λ̇t = 0 shifts upward.11
The shift of the locus ẇt = 0 depends on a value of λt .12 The results of diﬀerentiating the righthand side of (20) with respect to G imply that the graph of (20) shifts upward (downward)
] 1
[ αa
N
1−α+β
. We illustrate the two cases in Figure 2: in the ﬁrst case,when λt < (>) (1−α)aB
1[][ αaN ] 1
αaN
∗ <
1−α+β
1−α+β
λ∗ > (1−α)a,whereasinthesecondλ. Note that wt is a jump(1−α)aB
B
variable and λt is a predetermined variable. In both cases, the level of wt jumps up initiallyand thereafter monotonically increases to the new steady-state level, and λt decreases andapproaches the new steady state. These results allow us to create Figure 3.

10

Many empirical studies ﬁnd that public research can complement or substitute private R&D. See David etal. (2000) for an extensive survey.
11
The shift of locus λ̇t = 0 due to an increase in G is obtained from (21) as follows:B
1 + aaN
λ1−α+βt(ε − 1)θ
dwt=()2 > 0.
aB
dG λ̇t =0,dλt =0
ε1−G−Gλt1−α+βaN

12The shift of locus ẇt = 0 due to an increase in G is obtained from (20) as follows:
dwt
dG

=
ẇt =0,dλt =0

aN {1 + α(ε − 1)θ}αaN − (1 − α)aB λ1−α+β
t
{
}2 .
ε
αaN (1 − G) + (1 − α)aB Gλ1−α+β+ ρλ1−αttHence, the shift depends on the value of λt .

12

Figure 2: The eﬀects of an increase in G.

Figure 3: Dynamic paths of wt and λt after an increase in G.

13

Next, we examine the eﬀects of changes in G on the growth in the number of diﬀerentiatedtgoods, γtN ≡ ṄNt , considering the transitional dynamics to the new steady state by following

Judd’s (1982, 1985) method. Using linearized versions of the diﬀerential equations (18) and (19)around their steady-state values, we calculate eﬀects of marginal changes in G on the values of
wt and λt within the transitional dynamics to the new steady state. Taking into account theinitial condition, w0 = w∗ and λ0 = λ∗ , we show in Appendix A thatdw∗ dλ∗ −ϕt
dwt
=
+
νe ,
dG
dG
dG
(32)
) dλ∗
dλt (
= 1 − e−ϕt,dGdGwhere ν and ϕ are positive. From (12), (16), and (32), the eﬀect of changes in G on the valueof γtN within the transitional dynamics to the new steady state is as follows:{}dγtN
ε − 1 1 ( dw∗dλ∗ −ϕt )∗ α−1= aN (λ )−1 + θ
+νedG
ε (w∗ )2 dG
dG
(
)
) ∗
ε−1 1 (
∗ α−2
−ϕt dλ−(1 − α)aN (λ )
1−G−θ
1
−
e
.
ε w∗
dG

(33)

N

∞
In this analysis, we focus on the case in which 0 < G < Gg ; that is, dγ
dG > 0 holds. The initialdγ N

eﬀect of G on the growth in the number of the diﬀerentiated goods, dG0 , is as follows:{}ε − 1 1 dw0
dγ0N
∗ α−1
= aN (λ )−1 + θ.
dG
ε (w∗ )2 dG
The ﬁrst term represents a crowding-out eﬀect on the labor devoted to applied research, and0
the second represents the eﬀect of the demand for skilled labor on production. Because dw
dG is

dγ N

dγ N

positive, the sign of dG0 is not obvious. By using (32), we rearrange dG0 as follows:[]dγ0N
ε − 1 1 dw∗
ν dλ∗
∗ α−1
∗ α−1 ε − 1= aN (λ )−1 + θ
+a
(λ
)
θ
.
N
dG
ε (w∗ )2 dGε (w∗ )2 dG
|
{z
}
dL∗
R
dG

Equation (26) and ν > 0 imply that the ﬁnal term is negative. The expression in square brackets
dL∗

dL∗

is equivalent to dGR . As discussed in the previous subsection, dGR < 0 holds when G̃ ≤ G < 1;
dγ N

dγ N

in this case, dG0 < 0 holds. However, when 0 < G < G̃, the sign of dG0 is ambiguous becausedL∗R
dG > 0.
dγ N

In Figure 4, we illustrate the case in which dG0 is positive and that in which it is negative. As
shown, when the government increases G at time 0, the growth in the number of diﬀerentiatedgoods, γtN , initially jumps down or up and thereafter monotonically increases to the new steadystate level. Although the short-run growth eﬀect is ambiguous, the long-run eﬀect is certainly
positive because we focus on analyzing a speciﬁc range of the policy variable, 0 < G < Gg .
14

Figure 4: Dynamic paths of γtN after an increase in G.

5

Welfare eﬀects

5.1

Steady-state welfare

The main purpose of this paper’s analysis is to examine the implications of basic research on
household welfare. We ﬁrst investigate the welfare level in the steady state. Equations (4) and(6) yield
11

ε − 1 Ntε−1
N ε−1
=
.
Ct = tpt
ε z(wt , wtu )Substituting (9) and (17) into (34), we obtain
( )
[ (
]
∫ t
1
1
1
ε − 1 )θ 1−θ ε−1
log Ct = θ log+γτN dτ + log ax
θ L N0
.
wt
ε−1 0
ε

(34)

(35)

1
(
)θ 1−θ ε−1= 1. In order to restrict our attention
Without any loss of generality, we set ax ε−1
θ
L
N
0
ε

to welfare in the steady state, we assume that the economy is initially in the steady state. Hence,∫t
wt = w∗ and 0 γτN dτ = γ ∗ t hold. By using (1) and (35), the steady-state welfare level, U ∗ , canbe calculated by
( )
θ
1
1
U = log
+ 2
γ∗.
∗
ρ
w
ρ (ε − 1)∗(36)

We examine the relationship between U ∗ and G. Diﬀerentiating U ∗ with respect to G, we obtain
dU ∗
θ dw∗
1
dγ ∗
=− ∗
+ 2
.
dG
ρw dG
ρ (ε − 1) dG(37)Here, (27), (29), and (37) jointly imply
dU ∗
< 0,
dG G=Ĝ
15[)
f or all Ĝ ∈ Gg , 1 .

(38)

∗

When 0 < G < Gg , the sign of dU
dG is ambiguous. To investigate this sign as G → 0, we use(22), (23), (27), and (28) to obtain
] 1
1−α
aN
lim λ =,
G→0
(ε − 1)θρ{}(ε − 1)θ aN + aB (λ∗ )1−α+β
dw∗
lim
=
,
G→0 dG
εaN∗[

lim w∗ =G→0(ε − 1)θ
,
ε

dγ ∗
aN aB
lim
=
(λ∗ )α+β−1 .
G→0 dG(ε − 1)θρ(39)

By using these results and (37), we can show thatdU ∗
θ= − < 0.
G→0 dGρlim(40)

Equation (40) implies that the growth eﬀect of the second term in (37) is smaller than the ﬁrst
term’s wage eﬀect when G is suﬃciently small.
Next, we consider whether there exists a steady-state welfare-maximizing level of G. Notethat the relationship between γ ∗ and G follows an inverted-U curve and w∗ is increasing in G.
Thus, from (36), (38), and (40), there is a steady-state welfare-maximizing level of G if theabsolute value of the ﬁrst term in (36) is suﬃciently small and the value of the second termin (36) is suﬃciently large at an intermediate level of G. To derive this condition, we use thefollowing lemma (see Appendix C):

Lemma 1The steady-state wage rate for skilled labor, w∗ , and the steady-state growth rate, γ ∗ , are bothincreasing in aN , aB and decreasing in ρ.
Lemma 1 states that the steady-state skilled labor wage rate and growth rate (w∗ and γ ∗ )are high for all G ∈ (0, 1) if aN and aB are large and ρ is small. Note that log( w1∗ ) can become
negative if G is suﬃciently high.13 Therefore, if we impose the condition that log( w1∗ ) > 0 forG ∈ (0, Gg ), suﬃciently large aN and aB and small ρ imply that log( w1∗ ) is suﬃciently small
and γ ∗ is suﬃciently large at an intermediate level of G. As shown in Appendix D, the condition
that log( w1∗ ) > 0 for G ∈ (0, Gg ) is as follows:14{}
1 + (ε − 1)θ aB Gg (λ̃)1−α+β + (ε − 1)θρ(λ̃)1−α − aN (1 − Gg ) < 0,

(41)

13
From (22), we obtain limG→1 λ∗ = 0. This and (23) yield limG→1 w∗ = +∞. Because w∗ is increasing in G
and limG→0 w∗ = (ε − 1)θ/ε < 1, w∗ > 1 holds if G is suﬃciently large.
14
The wage rate for skilled labor might be lower than that for unskilled labor. From the above discussion, thecondition for wt > wu is limG→0 w∗ > wu . By using (17) and (39), we impose the following condition: L > 1−θ.
θ16where

[

aN
λ̃ ≡
ρ

{

1 + (ε − 1)θ
− (1 − Gg )
ε

}] 1

1−α

.

Under condition (41), suﬃciently large aN and aB and small ρ imply that there is a steadystate welfare-maximizing level of G.15 To demonstrate this clearly, we employ some numericalexamples. The upper-left panel of Figure 5 corresponds to the case in which aN is suﬃciently
large; this shows that there is a steady-state welfare-maximizing level of G. The upper-rightpanel of Figure 5 corresponds to the case in which aN is not large. In this case, U ∗ is increasingin G at the intermediate level of G, but, as the value of the second term in (36) is small, G = 0
maximizes steady-state welfare. The lower panel of Figure 5 corresponds to the case in which
aN is suﬃciently small, indicating that U ∗ is decreasing in G. That is, G = 0 maximizes steadystate welfare.16 In addition, as shown in Appendix E, we obtain similar results when we vary
the values of aB and ρ while holding the other values ﬁxed.
Let us deﬁne Gw by the steady-state welfare-maximizing level of G and use this to compare
the steady-state welfare-maximizing level of G with its steady-state growth-maximizing level.
From the deﬁnition of Gw and (38), it is easy to conﬁrm that the steady-state welfare-maximizing
level of G is lower than the steady-state growth-maximizing level, as summarized in the following
proposition.

Proposition 1
Suppose that aN and aB are suﬃciently large and ρ is suﬃciently small. Setting G = Gw
maximizes steady-state welfare. In addition, the steady-state welfare-maximizing level of G is
below the steady-state growth-maximizing level.

As mentioned above, an increase in G raises the wage rate for skilled labor; as a result,the price of goods increases, reducing household consumption. Therefore, there is a tradeoﬀ between the negative eﬀect on household consumption and the positive growth eﬀect if
0 < G < Gg . When the government increases G to maximize the growth rate, this depresses
household consumption; the steady-state welfare-maximizing level of G is thus below the steadystate growth-maximizing level.
15
The eﬀects of ε and θ on U ∗ are ambiguous because w∗ is increasing in ε and θ while γ ∗ is decreasing in εand θ. This derivation is shown in Appendix C.
16
In the cases in which aN = 0.057 and aN = 0.055, U ∗ is decreasing in G for G > 0.15. To clearly demonstrate
the relationship between U ∗ and G, we omit the range in which G > 0.15.

17

Figure 5: The relationship between U ∗ and G. The parameter values are
aB = 0.2, ρ = 0.05, ε = 4, α = 0.4, β = 0.6, and θ = 0.3.

5.2

Welfare eﬀects of policy changes

In this subsection, we examine the welfare eﬀects of marginal changes in G using Judd’s (1982,
1985) method. Suppose that the economy is initially in the steady state. When the government
increases G at time 0, the economy undergoes transitional dynamics, eventually converging to
the new steady state. From (1) and (35), the overall eﬀects of marginal changes in G on thewelfare are given by
dU0
=−
dG

∫ ∞
0

θ dwt
1
e−ρt ∗
dt +
w dG
ε−1

18

∫ ∞
e
0

−ρt

∫ t

dγτN
dτ dt.
0 dG(42)

Equations (32), (33), and (42) imply
θ dw∗
θν
dλ∗
dU0
=− ∗
−
dG
ρw dG
(ρ + ϕ)w∗ dG[ {]}aN (λ∗ )α−1 1
ε − 1 1 dw∗
ε−1 1
ν dλ∗
+
−1 + θ+θ(43)(ε − 1)ρ ρ
ε (w∗ )2 dGε (w∗ )2 ρ + ϕ dG)(
(1 − α)aN ϕ(λ∗ )α−2
ε − 1 1 dλ∗
− 2
1−G−θ
.
ρ (ε − 1)(ρ + ϕ)ε w∗ dG0
By using (43), we evaluate dU
dG when the government initially sets G to Gw . As shown inAppendix F, we obtaindU0< 0.
dG G=Gw
In summary, we can state the following proposition:

Proposition 2
Suppose that aN and aB are suﬃciently large, ρ is suﬃciently small, and the economy is initiallyin the steady state. If G is set to Gw , a decrease in G marginally increases the welfare ofhouseholds.

From the discussion in subsection 4.2, when the government increases G at time 0, the wage
rate for skilled labor, wt , initially jumps up and thereafter monotonically increases to the newsteady-state level. Meanwhile, growth in the number of diﬀerentiated goods, γtN , jumps down or
up initially and thereafter monotonically increases to the new steady-state level.17 As discussed
above, an increase in the wage rate for skilled labor reduces household consumption. Thus,
there is a trade-oﬀ between the negative eﬀect on household consumption and the positive longrun growth eﬀect. Proposition 2 states that if G is equal to its steady-state welfare-maximizing
level, the former eﬀect overwhelms the latter eﬀect. Proposition 2 also states that the optimal
constant level of G is lower than the level that maximizes steady-state welfare.

17
We compare Gw to G̃ using the above example numerical values (aN = 0.1, aB = 0.2, ρ = 0.05, ε = 4,
α = 0.4, β = 0.6, and θ = 0.3). With these parameter values, Gw ≈ 0.193 and G̃ ≈ 0.182; thus, Gw > G̃ holds.
When the government increases G at time 0, the growth in the number of diﬀerentiated goods, γtN , jumps down
initially and thereafter monotonically increases to the new steady-state level.

196Conclusion

In this study, we developed a variety expansion growth model that integrated applied and basic
research sectors, seeking chieﬂy to examine the eﬀects of basic research on household welfare.
The analysis derived two key results. First, the steady-state welfare-maximizing level of basic
research is below the steady-state growth-maximizing level. Second, a reduction in the levelof basic research raises welfare if the level of basic research is set to the steady-state welfaremaximizing level.
We see several interesting directions for future research. First, this study used a ﬁrstgeneration R&D-based growth model that exhibits scale eﬀects; that is, an increase in the sizeof the labor force raises the growth rate. However, Jones (1995) ﬁnds that scale eﬀects are
not supported by empirical evidence.18 In future research, it would be interesting to consider
non-scale growth models. Second, the assumption that the supply of skilled and that of unskilled labor are exogenous seems unrealistic. To address this, future research could incorporate
endogenous skill acquisition following Dinopoulos and Segerstrom (1999).

AppendixA.

Local stability

We examine the local stability at the steady state. Approximating (18) and (19) linearly in theneighborhood of the steady state, we obtain
)(
)
( ) (
ẇt
Jww Jwλwt − w ∗.
=Jλw Jλλλt − λ∗λ̇t(A.1)

Here, Jij (i, j = w, λ) denotes entities in the Jacobian matrix of this system:} (λ∗ )α−1aN {1 + α(ε − 1)θ
> 0,
ε
w∗
{
} λ∗
Jwλ = (1 − α) ρ + (1 − α + β)aB G(λ∗ )β
> 0,
w∗
∗
α
ε−1
(λ )
Jλw = θaN ∗ 2 > 0,
ε
(w )
)
(
ε−1 1
(λ∗ )α−1 < 0.
Jλλ = −(1 − α + β)aN 1 − G − θ
ε w∗

Jww =

The eigenvalues of the Jacobian matrix, J, are deﬁned as χi (i = 1, 2). Here, χ1 and χ2 are theroots of the characteristic equation, χ2 − (Jww + Jλλ )χ + Jww Jλλ − Jwλ Jλw = 0. From the sign18

See Jones (1995) for a more detailed discussion of scale eﬀects in R&D-based growth models.

20

of Jij , we obtain Jww Jλλ − Jwλ Jλw < 0. We then have√Jww + Jλλ + (Jww + Jλλ )2 − 4(Jww Jλλ − Jwλ Jλw )
χ1 =
> 0,
2
√
Jww + Jλλ − (Jww + Jλλ )2 − 4(Jww Jλλ − Jwλ Jλw )
χ2 =
< 0.
2
Note that wt is a jump variable and λt is a predetermined variable. Thus, the steady state islocally saddle-point stable.
Next, in order to calculate the eﬀects of marginal changes in G on the values of wt andλt within the transitional dynamics to the new steady state, we solve the linear diﬀerential
equations (A.1) as follows:
wt = w∗ − (λ0 − λ∗ )νe−ϕt ,λt = λ∗ + (λ0 − λ∗ )e−ϕt ,where ν ≡ −(χ2 − Jλλ )/Jλw and ϕ ≡ −χ2 . Diﬀerentiating wt and λt with respect to G yields
dwt
dw∗ dλ∗ −ϕtd ( −ϕt )=+νe− (λ0 − λ∗ )
νe
,
dG
dG
dG
dG
) dλ∗
dλt (
d ( −ϕt )= 1 − e−ϕt
+ (λ0 − λ∗ )
e
.
dG
dG
dG

(A.2)
(A.3)

Here, we assume that the economy is initially in the steady state (i.e., λ0 = λ∗ ). As a result,
the third term in (A.2) and the second term in (A.3) become zero.
Finally, we examine the sign of ν. From the deﬁnitions of ν and χ2 , we obtain√
Jww − Jλλ − (Jww + Jλλ )2 − 4(Jww Jλλ − Jwλ Jλw )
ν=−
.
2Jλw
Jww > 0 and Jλλ < 0 imply that Jww − Jλλ > 0. Hence, we calculate the following diﬀerence:{}(Jww − Jλλ )2 − (Jww + Jλλ )2 − 4(Jww Jλλ − Jwλ Jλw ) = −4Jwλ Jλw < 0.
Thus, ν > 0 holds.

B.

Relation between dL∗R /dG and G

Using (27), we obtaindL∗R
aB ∗ β−(1 − α + β)aN G + (1 − α)(ε − 1)θρ(λ∗ )1−α
=
.
(λ )
dG
aN
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ

21

(
)
Note that λ∗ depends on G see equation (22) . Hence, we deﬁne a function as Λ(G) =dL∗−(1 − α + β)aN G + (1 − α)(ε − 1)θρ(λ∗ )1−α . The sign of dGR determines the sign of Λ(G). (22)
[ a
] 1
N
1−α
implies that limG→0 λ∗ = (ε−1)θρand limG→1 λ∗ = 0. By using these results, we obtainlim Λ(G) = (1 − α)aN > 0 and

G→0lim Λ(G) = −(1 − α + β)aN < 0.

G→1

From (26), Λ(G) is decreasing in G. Thus, Λ(G) is positive (negative) when G is small (large).
Here, G̃ is deﬁned as Λ(G̃) = 0. As a result, we obtain the following relation:dL∗R
R 0 ⇔ G Q G̃.
dGC.

Proof of Lemma 1

By using (22), (23), (24), and (25), we diﬀerentiate λ∗ , w∗ , and γ ∗ with respect to ρ as follows:dλ∗(ε − 1)θλ∗
=−
< 0,
dρ
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ
εaB G(w∗ )2
dλ∗
dw∗
=
(1 − α + β)(λ∗ )−α+β< 0,dρ(ε − 1)θaN
dρ
dγ ∗
dλ∗
= βaB G(λ∗ )β−1
< 0,
dρ
dρ
With regard to aN , aB , ε, and θ, we obtain the following results in a similar way:dλ∗(1 − G)(λ∗ )α
=
> 0,
daN
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ
εaB (w∗ )2 (λ∗ )ββ(ε − 1)θρ(λ∗ )1−α
dw∗
=
> 0,
daN
(ε − 1)θa2N (1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρdγ ∗dλ∗= βaB G(λ∗ )β−1
> 0,
daN
daN
{1 + (ε − 1)θ}G(λ∗ )1+β
dλ∗
=−
< 0,
daB
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρdw∗
εG(w∗ )2 (λ∗ )1−α+β(1 − α)(ε − 1)θρ
> 0,
=
daB
(ε − 1)θaN(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρdγ ∗(1 − α){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ
< 0,
= G(λ∗ )βdaB(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρdλ∗
θaB G(λ∗ )1+β + θρλ∗
=−
< 0,
dε
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρ
{
}[
]
dw∗
(w∗ )2 (λ∗ )1−α aB G(λ∗ )β + ρ (1 − α + β)(1 − θ)aB G(λ∗ )β + (1 − α)(ε − 1)θρ
=
> 0,
dε
(ε − 1)θaN(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρdγ ∗dλ∗= βaB G(λ∗ )β−1
< 0.
dε
dε
22

dλ∗
(ε − 1)aB G(λ∗ )1+β + (ε − 1)ρλ∗
=−
< 0,
dθ
(1 − α + β){1 + (ε − 1)θ}aB G(λ∗ )β + (1 − α)(ε − 1)θρε(ε − 1)(w∗ )2 ((1 − α)ερ(w∗ )2dw∗
ρ ∗ 1−α )
dλ∗
=
(λ )
−
(λ∗ )−α
> 0,
1
−
G
+
2
dθ
{1 + (ε − 1)θ}aN
{1 + (ε − 1)θ}aN
dθ
dγ ∗
dλ∗
= βaB G(λ∗ )β−1
< 0.
dθ
dθ
Therefore, the wage rate for skilled labor, w∗ , is increasing in aN , aB , ε, and θ but decreasingin ρ. The growth rate, γ ∗ , is decreasing in ρ, ε, and θ but increasing in aN and aB .

D.

Derivation of condition (41)log( w1∗ ) is positive if w∗ < 1 holds. By using (25), the condition w∗ < 1 is as follows:
λ∗ >

[

aN
ρ

{

1 + (ε − 1)θ
− (1 − G)
ε

}] 1

1−α

.

(D.1)

From (26), the left-hand side of (D.1) is decreasing in G and the right-hand side of (D.1) isincreasing in G. Thus, to ensure that log( w1∗ ) is positive for G ∈ (0, Gg ), it is suﬃcient that(D.1) holds at G = Gg , that is,[aN
λ G=Gg >
ρ
∗

{

}] 1
1−α
1 + (ε − 1)θ
− (1 − Gg )
≡ λ̃.
ε

(D.2)

By using (22), the condition (D.2) is as follows:{}
1 + (ε − 1)θ aB Gg (λ̃)1−α+β + (ε − 1)θρ(λ̃)1−α − aN (1 − Gg ) < 0.

E.

Numerical examples (steady-state welfare)

For the baseline parameter values, we choose aN = 0.1, aB = 0.2, ρ = 0.05, ε = 4, α = 0.4,
β = 0.6, and θ = 0.3 (this corresponds to the upper-left panel of Figure 5). To demonstrate
the relationship between U ∗ and G, we vary the value of aB (ρ) while holding the others ﬁxed
in Figure E.1 (E.2). In both ﬁgures, properties of the relation between U ∗ and G are the sameas in Figure 5. From these results, we conﬁrm that there is a steady-state welfare-maximizing
level of G if aN and aB are suﬃciently large and ρ is suﬃciently small.

23

Figure E.1: The parameter values are aN = 0.1, ρ = 0.05, ε = 4,
α = 0.4, β = 0.6, and θ = 0.3.

Figure E.2: The parameter values are aN = 0.1, aB = 0.2, ε = 4,
α = 0.4, β = 0.6, and θ = 0.3.

24

F.

Proof of Proposition 2∗

The deﬁnition of Gw implies that dU
dG G=Gw = 0 holds. By using (30), (31), and (37), we obtain
{
}
θ dw∗
ε − 1 1 dw∗
aN (λ∗ )α−1−
−1 + θ
+ρw∗ dG G=Gw ρ2 (ε − 1)ε (w∗ )2 dG G=Gw()
(1 − α)aN (λ∗ )α−2
ε − 1 1 dλ∗
=
1−G−θ
.
ρ2 (ε − 1)ε w∗ dG G=Gw

(F.1)

Substituting (F.1) into (43) yields{}θν(λ∗ )α−1 dλ∗dU0=−ερ + aNdG G=Gw ερ(ρ + ϕ)w∗w∗
dG G=Gw()(1 − α)aN (λ∗ )α−2
ε − 1 1 dλ∗
+
1−G−θ
.
ρ(ε − 1)(ρ + ϕ)ε w∗ dG G=Gw

(F.2)

From (21) and (22), we obtain−ερ + aN(λ∗ )α−1= εaB G(λ∗ )β .
w∗

(F.3)

By using (26), (F.3), ϕ > 0, and ν > 0, (F.2) becomesdU0< 0.
dG G=Gw

References[1] Arnold, L.G., 1997. Basic and applied research. FinanzArchiv: Public Finance Analysis 54,
169-186.
[2] Chu, C., Cozzi, G., Galli, S., 2012. Does intellectual monopoly stimulate or stiﬂe innovation? European Economic Review 56, 727-746.
[3] Chu, C., Furukawa, Y., 2013. Patentability and knowledge spillovers of basic R&D. Southern Economic Journal 79, 928-945.
[4] Cohen, W.M., Nelson, R.R., Walsh, J.P., 2002. Links and impacts: the inﬂuence of publicresearch on industrial R&D. Management Science 48, 1-23.
[5] Cozzi, G., Galli, S. 2009. Science-based R&D in Schumpeterian growth. Scottish Journalof Political Economy, 56, 474-491.
[6] Cozzi, G., Galli, S., 2013. Privatization of knowledge: Did the U.S. get it right? Economics
Working Paper Series 1307, University of St. Gallen, School of Economics and Political
Science.
[7] Cozzi, G., Galli, S., 2014. Sequential R&D and blocking patents in the dynamics of growth.
Journal of Economic Growth 19, 183-219.
25

[8] David, P., Hall, B., Toole, A., 2000. Is public R&D a complement or substitute for private
R&D? A review of the econometric evidence. Research Policy 29, 497-529.
[9] Dinopoulos, E., Segerstrom, S., 1999. A Schumpeterian model of protection and relative
wages. American Economic Review 89, 450-472.
[10] Gersbach, H., Schneider, M., 2015. On the global supply of basic research. Journal of
Monetary Economics forthcoming.
[11] Gersbach, H., Schneider, M., Schneller, O. 2013. Basic research, openness, and convergence.
Journal of Economic Growth 18, 33-68.
[12] Gersbach, H., Sorger, G., Amon, C., 2009. Hierarchical growth: Basic and applied research.
CER-ETH Working Papers 118, CER-ETH-Center of Economic Research at ETH Zurich.
[13] Griliches, Z., 1986. Productivity, R&D, and basic research at the ﬁrm level in the 1970’s.
American Economic Review 79, 957-970.
[14] Grossman, G.M., Helpman, E., 1991. Innovation and growth in the global economy. MIT
Press, Cambridge, MA.
[15] Jaﬀe, A.B., 1989. Real eﬀects of academic research. American Economic Review 76, 141154.
[16] Jones, C.I., 1995. R&D-based models of economic growth. Journal of Political Economy
103, 759-784.
[17] Judd, K.L., 1982. An alternative to steady-state analysis in perfect foresight models. Economics Letters 10, 55-59.
[18] Judd, K.L., 1985. Short-run analysis of ﬁscal policy in a simple perfect foresight model.
Journal of Political Economy 93, 298-319.
[19] Konishi, K., 2013. Public research spending in an endogenous growth model. Discussion
Papers in Economics and Business 13-26, Osaka University.
[20] Mansﬁeld, E., 1998. Academic research and industrial innovation: An update of empiricalﬁndings. Research Policy 26, 773-776.
[21] National Science Foundation, 2011. Research and development in industry:
2006-07. Detailed Statistical Tables NSF 11-301. Arlington, VA. Available at
http://www.nsf.gov/statistics/nsf11301/.
[22] Park, W.G., 1998. A theoretical model of government research and growth. Journal of
Economic Behavior and Organization 34, 69-85.

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