The Inhibitory Effect of Population
Ageing on Technical Progress

Hideo Noda
Research Group of Economics and Management
No. 2010-E02
2010.8YAMAGATA UNIVERSITY FLSS DISCUSSION PAPER SERIESThe Inhibitory Eﬀect of Population Ageing on
Technical Progress∗
Hideo Noda†
Discussion Paper 2010-E02
August 2010

Abstract
This article aims to clarify the relationship between population ageing andtechnical progress. Our discussion is carried out in terms of a model of endogenous growth with expanding product variety. The model implies that therate of innovation will decline with population ageing in the long term. Thisphenomenon can be interpreted as ageing having an inhibitory eﬀect on technical progress and thus economic growth. In addition, we ﬁnd that raising the
mandatory retirement age has a positive eﬀect on technical progress. This suggests that encouraging the employment of elderly people is important from theperspective of promoting economic growth.
Keywords: Knowledge-based Economy; Population Ageing; R&D; Technical
Progress; Economic Growth
JEL classification: E60; J10; J24; O31; O38∗

An earlier version of this paper was presented at the 67th Annual Meeting of the Japan Economic
Policy Association, May 30, 2010, Kyoto. I would like to thank Daisuke Ikazaki, Hiroyuki Kawanobe,and Haruki Niwa for their constructive comments and suggestions. This research was partially
supported by the Grant-in-Aid for Young Scientists (B) (21730218) from the Ministry of Education,
Culture, Sports, Science and Technology.
†
Faculty of Literature and Social Sciences, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata 990-8560, Japan, (e-mail) noda@human.kj.yamagata-u.ac.jp1

1

Introduction

Many developed countries are facing the inexorable progress of population ageing.
The level of Japan’s ageing, in particular, is much higher than that of other developed countries. Moreover, it has been progressing rapidly. Such a demographic
phenomenon raises the question: What eﬀects does population ageing have on thetrends of the future macro economy? Although various arguments, both optimistic
and pessimistic, have been advanced in this regard, no clear consensus has beenreached to date. Therefore, the ageing of population is a long-term major concern
for economists and policy makers. Incidentally, as exempliﬁed using terms such as
knowledge-based economy, the economic activities of major industries in developed
countries are becoming increasingly knowledge-intensive today. A salient featureof a knowledge-based economy is that research and development (R&D) activities
in response to perceived proﬁt opportunities create innovation, which is accompanied by the accumulation of technological knowledge that has an important role inlong-term economic growth.
Considering the social and economic conditions described above, population ageing and innovation are important concepts that cannot be ignored in the analyses
of modern economic policies. Nevertheless, theoretical studies on the link between
population ageing and innovation are extremely rare. Earlier economic analysesof ageing are dominated by themes related to social security, such as public pensions and health insurance. Social security is unquestionably relevant to ageing.
However, the ageing of a population and maturing of a knowledge-based economy
progress simultaneously, and gaining insight into the eﬀect of ageing on innovationand thus economic growth is therefore crucial to eﬀective public policy research.
The purpose of this article is to clarify the eﬀect of population ageing on the longterm performance of a knowledge-based economy in the context of a growth modelwith endogenous innovation. That is, this article is concerned with the relationshipbetween population ageing and economic growth, not social security problems.
Recent works from an endogenous growth theory perspective can be classiﬁedroughly into two types. The ﬁrst type focuses on the role of innovation in economic
growth. Futagami and Nakajima (2001) used an extended life cycle growth model2

to show that population ageing is not necessarily a negative factor for economic
growth. In their model, the engine of long-term growth is learning-by-doing, which
can be interpreted as a kind of incremental improvement innovation. Also, Futagami, Iwaisako and Nakajima (2002) used a model of product innovation throughindustrial R&D to conﬁrm that population ageing would have a negative eﬀect onmacroeconomic growth. The second type stresses human capital. For example,
Gruescu (2007, Ch.7) extended the growth model of Lucas (1988) emphasising theworkers’ skill and its externality. As a result, Gruescu’s (2007, Ch.7) model predicts
lower economic growth in an economy if the population is ageing.
The marked diﬀerence between this article and the existing literature is that it
considers the roles of both human capital and innovation. As OECD (2001) and
Psarras (2007) argue, in recent years an improvement in workers’ skills has been
demanded more than ever in developed countries. Therefore, the importance ofhuman capital formation has been increasing further as the economic activities ofindustries become more knowledge-intensive. Such a tendency suggests the necessity of building a framework that treats human capital in addition to innovation
as endogenous variables of the model and reﬂects the reality of a knowledge-based
economy. However, most earlier studies emphasise either innovation or human capital, not both. As a result, the modelling of those earlier studies inadequately reﬂects
the essential characteristics of a knowledge-based economy. We therefore conclude
that the results of the analysis presented in the relevant literature described above
lack suﬃcient persuasiveness.
Our model is based on that of Grossman and Helpman (1991, Ch.5), even though
they did not examine the relationship between population ageing and technicalchange. By extending their model we present a manageable framework for analysing
macroeconomic eﬀects of population ageing. A main ﬁnding derived from our modelis that the rate of innovation will decline with population ageing in the long term.
This phenomenon can be interpreted as ageing having an inhibitory eﬀect on technical progress and therefore also on economic growth in a knowledge-based economy.
In addition, we ﬁnd that raising the mandatory retirement age has a positive eﬀect
on technical progress. Thus, the model implies that encouraging the employment of3

elderly people is important from the viewpoint of promoting economic growth.
The article is organised as follows. In Section 2 we introduce a model of economic
growth through innovation and analyse the relationship between population ageing
and technical progress. In Section 3 we examine the policy implications derived fromthe model. Finally, in Section 4 the main results and conclusions are presented.

2

The Model

2.1

Households

Let us start by describing the setup of the model for households. We consider
a representative agent who lives for a ﬁnite period of time T . In this economy,
individuals in age groups ranging from 0 to T are distributed continuously at eachpoint in time; the density of individuals in each age group is N/T . We also assume
that the number of births is the same as the number of deaths at each point intime. Thereby, the total population N is constant. Based on these assumptions,representing age by q, the density function f (q) of age is:{ N(0 ≤ q ≤ T ),Tf (q) =
0 (otherwise).
Two worker types are considered. One is an individual equipped with skills
to engage in R&D activities. The other is an individual without such skills. In
the model, individuals of the ﬁrst type are called skilled workers, and those of thesecond type are called unskilled workers. For convenience, the set of the worker
types is expressed as B, including skilled workers as h, and unskilled workers as l,that is, B = {h, l}. We also assume that all individuals retire mandatorily at theage of Z(< T ). They work until the mandatory retirement age of Z irrespectiveof whether they are skilled or unskilled. This can be interpreted as all individuals
retiring at the same age Z. We treat N , T , and Z as parameters for the model.
In the model, we assume a closed economy, whose industries comprise two sectors, the nondurable consumer goods sector and the R&D sector. It is assumed that
unskilled workers supply labour to the consumer goods sector and that skilled workers supply labour to the R&D sector. We regard unskilled labour as the numéraire,so that the wage rate paid for a unit of unskilled labour is normalised to 1. Skilledworkers are paid for embodied human capital at wage rate w(t) at time t.

4

We analyse the consumer behaviour of households based on the settings above.
People who are born at time point κ are called generation κ. The lifetime utility
Ui (κ) of an individual of type i from generation κ is given by∫ κ+TUi (κ) =e−ρ(t−κ) ln Di (κ, t)dt.

(1)

κThe parameter ρ denotes the subjective discount rate, and it is assumed that ρ > 0.
In addition, ln Di (κ, t) is the instantaneous utility of an individual of type i fromgeneration κ at time t. The type of consumer goods is assumed to be a continuous
quantity. The consumer goods that are available at time t are speciﬁed as the interval[0, A(t)]. Thus, A(t) denotes the measure of consumer goods invented before time t.
Households recognise each of the consumer goods as diﬀerent items, and so productdiﬀerentiation is assumed. The consumption index, Di (κ, t), is formulated as]1[ ∫ A(t)αxij (κ, t)α dj ,Di (κ, t) =(2)0where xij (κ, t) is the amount of the j-th product consumed by an individual of typei from generation κ at time t. In addition, the parameter α takes values between 0
and 1. The elasticity of substitution between any two consumer goods is 1/(1 − α).
Next, we represent the budget constraints that households face. We assume
that households may freely lend and borrow at an instantaneous interest rate ofr(t), and that no assets or debts remain at the time of death. For an individual oftype i from generation κ, total expenditure is expressed as Ei (κ, t), and the sum of
the discounted present values of income ﬂow acquired is expressed as Wi (κ), so theintertemporal budget constraint of an individual of type i from generation κ is given
by

∫ κ+T

∫t

e− κ r(s)ds Ei (κ, t)dt ≤ Wi (κ).

(3)κ

The constrained optimisation problem for an individual of type i from generation
κ can be considered using the following two steps. The ﬁrst step is to ﬁnd thequantity of each type of goods demanded that will maximise the instantaneous
utility for given expenditure and prices. In other words, a household of type i fromgeneration κ will solve the following problem:[ ∫ A(t)]1αmaxxij (κ, t)α dj ,05∫ A(t)s.t.

pj (t)xij (κ, t)dj ≤ Ei (κ, t).

0Solving this problem yields
1

pj (t) α−1 Ei (κ, t)xij (κ, t) = ∫ A(t)pm (t)

α
α−1

.

(4)

dm

0

We express the number of households of type i from generation κ in the subjective equilibrium as Mi (κ, t). On the assumption that each age group is distributed∑
uniformly at N/T , the relation i∈B Mi (κ, t) = N/T holds for all generations at
each point in time. The aggregate demand of households from generation κ for thej-th product is xj (κ, t), and the aggregate expenditure of households from generation
κ is E(κ, t). That is, xj (κ, t) and E(κ, t) are respectively given by∑xj (κ, t) =Mi (κ, t)xij (κ, t),(5)

Mi (κ, t)Ei (κ, t).

(6)

i∈B

∑

E(κ, t) =i∈B

Using Eqs. (4)-(6), the aggregate demand of households from generation κ for thej-th product, xj (κ, t), can be derived as1

pj (t) α−1 E(κ, t)xj (κ, t) = ∫ A(t)pm (t)

α
α−1

.

dm

0

When we express the aggregate demand for the j-th product as xj (t) and the aggregate expenditure of households in the economy as E(t), we obtain∫ t
xj (t) =xj (κ, t)dκt−T

pj (t)1α−1∫ tE(κ, t)dκt−T∫ A(t)=αpm (t) α−1 dm
01=

pj (t) α−1 E(t)∫ A(t)pm (t)
where E(t) ≡

∫t

α
α−1

,

(7)

dm

0

t−T E(κ, t)dκ．Note that the price elasticity of the demand for thej-th product is 1/(1 − α).

6The second stage of the problem is to ﬁnd the path of expenditure that maximises
the lifetime utility in Eq. (1), subject to the intertemporal budget constraint in Eq.
(3). Substitution from Eq. (4) into Eq. (2) gives
Di (κ, t) = [ ∫ A(t)Ei (κ, t)
pj (t)αα−1] α−1 .
dj

(8)

α

0

To simplify this expression, the denominator in Eq. (8) is denoted by PD (t), which
can be regarded as a kind of price index. That is,] α−1[ ∫ A(t)
α
α
α−1
dj
.
PD (t) ≡pj (t)
0

Furthermore, we take the natural logarithms of both sides of Eq. (8):ln Di (κ, t) = ln Ei (κ, t) − ln PD (t).
The optimisation problem at the second stage, therefore, is
∫ κ+T
max
e−ρ(t−κ) [ln Ei (κ, t) − ln PD (t)]dt,
κ
∫ κ+T
∫t
s.t.
e− κ r(s)ds Ei (κ, t)dt ≤ Wi (κ).
κ

As a result, we ﬁnd that the expenditure growth rate is given bydEi (κ, t)/dt= r(t) − ρ.
Ei (κ, t)

2.2

(9)

Producers of Consumer Goods

In the consumer goods market, there are numerous ﬁrms, and each ﬁrm leaves theeﬀect of decision making on other ﬁrms out of consideration. We assume that all
ﬁrms in the consumer goods sector are able to produce goods by purchasing patentrights to the exclusive use of the product technology from the R&D ﬁrms. As
described in Subsection 2.1, various consumer goods supplied to the market havephysical diﬀerences and consumers recognise such diﬀerences. Under this productdiﬀerentiation, each ﬁrm producing a speciﬁc consumer good has some control overits own product price. Accordingly, the structure of the consumer goods market ischaracterised by monopolistic competition.

7

The input of unskilled labour of the ﬁrm that produces the j-th product (calledﬁrm j from here on) is Lj . The following relation is assumed to hold between theamount of the j-th product produced and the input of unskilled labour of ﬁrm j:
xj =

1
Lj ,
ζ

(10)

where ζ(> α) is a positive constant. Evidently, the smaller the value of ζ, the higher
the productivity of unskilled labour of ﬁrm j. Because the wage rate of unskilled
labour is 1, the cost of input of unskilled labour is 1 × Lj = Lj . Using Eq. (10), theproﬁt πj of ﬁrm j is given byπj (t) = pj (t)xj (t) − Lj (t)= [pj (t) − ζ]xj (t).

(11)

The demand function for the product produced by ﬁrm j is given by Eq. (7).
Substituting Eq. (7) into Eq. (11) yields
1

pj (t) α−1 E(t)πj (t) = [pj (t) − ζ] ∫ A(t)pm (t)

α
α−1

.

(12)

dm

0

The price that maximises the proﬁt of ﬁrm j is given by
pj (t) =

ζ
≡ p.
α

(13)

Apparently, the price ζ/α exceeds the marginal cost ζ. Equation (13) implies
that the markup ratio is (1 − α)/α. If we substitute Eq. (13) into Eq. (7), then thequantity of the j-th product demanded isxj (t) =αE(t)≡ x(t).
ζA(t)

(14)

Furthermore, substitution of Eqs. (13) and (14) into Eq. (11) leads toπj (t) =( ζ − α ) E(t)ζA(t)≡ π(t).

(15)

Let Vj (t) be the discounted present value of the proﬁt ﬂow gained by ﬁrm j at timet. That is, Vj (t) is given by∫ ∞Vj (t) =∫τ

e− t r(ι)dι π(τ )dτ ≡ V (t).

t

8

2.3

R&D Firms

Suppose that R&D ﬁrms are continuously distributed in an interval [0, 1]. We consider the consumer goods design developer and consumer goods producer to be separate agents. Without considering the uncertainty in R&D activities, it is assumed
that new technology is deﬁnitely created through the input of human capital. Thus,
we use a deterministic framework for the invention of new products. The setup of
the R&D sector in our model is similar to those in Romer (1990) and Grossman andHelpman (1991, Ch.3).
We assume that if an R&D ﬁrm f ∈ [0, 1] inputs Hf units of human capital, thenit develops new product designs of ∆af during the period ∆t. More speciﬁcally, therelation ∆af = λΩHf ∆t holds. Here Ω denotes public knowledge capital and λ is apositive constant. Knowledge capital refers to the set of technological informationthat has been accumulated through R&D activities. It should be noted that thecontribution of public knowledge capital (Ω) to the creation of new product designs(∆af ) reﬂects the technological spillover by partial non-excludability of knowledge
capital.
Existing technological information oﬀers wide applicability and helps productplanners and product managers during the development of new product concepts.
Such explicit knowledge may diﬀuse widely across industries and leads to the formation of public knowledge capital over time. The accumulation of knowledge capitalfurther contributes to the creation of new product design. In fact, even if technology
is patented, the appropriability of technology through a patent is not complete, andtechnological spillover is commonly observed (Odagiri, 2001). A schematic representation of this innovation process is presented in Figure 1.

R&D sector

Innovation

Human capital

Knowledge capital

Figure 1: The Innovation Process

9The number of new technologies created by R&D ﬁrm f at time t is expressed
as follows:ȧf (t) = λΩ(t)Hf (t),(16)

where an overdot represents a time derivative. The sum total of the number of new∫1
technologies created by each R&D ﬁrm at time t, that is, 0 ȧf (τ )df , corresponds
to the number of new technologies developed in the macro economy, Ȧ(τ ). If we∫t ∫1
assume that limt→−∞ A(t) = 0, then A(t) = −∞ 0 ȧf (τ )df dτ holds.
Grossman and Helpman (1991, Ch.3) considered that the variety of consumer
goods reﬂects the cumulative experience of R&D and regarded it as a proxy variablefor knowledge capital. We follow them by assuming Ω(t) = A(t). Therefore, Eq.
(16) can be rewritten asȧf (t) = λA(t)Hf (t).

(17)

An R&D production function as described in Eq. (17) is often called a knowledgedriven speciﬁcation.
Suppose that an inventor of a new technology is granted a patent immediatelyby the government and that the duration of the patent is inﬁnite. We simplify the
analysis by assuming that the cost of maintaining the patent is zero. Each R&D ﬁrm
earns revenue by selling its patents to an agent attempting to enter the consumer
goods industry.
Recall that, in the subjective equilibrium, the discounted present value of the
proﬁt ﬂow for all ﬁrms producing consumer goods is equal to V (t). Consequently,
the patent price in the general equilibrium of an arbitrary R&D ﬁrm is equal to V (t).
We prove that the patent price of an arbitrary R&D ﬁrm equals V (t) by following
the argument of Jones (2002). Assume a situation in which anyone who attempts toenter the consumer goods market can participate in an English auction for patents.
Let us imagine that a patent of an R&D ﬁrm f is being auctioned. If the starting
price is higher than V (t), then there will be no incentive for anyone to bid. The
reason is that purchasing the patent and producing the consumer goods would only
result in a net loss. If the bid price of a participant is below V (t), then someone will
try to bid higher and eventually the bid price will be raised to V (t). Thus, when10

we consider the equilibrium in which the patent of the R&D ﬁrm f is sold and thenew patent holder produces the consumer good, the price of the patent of the R&D
ﬁrm f is expected to be V (t). This logic evidently applies to the transfer price of
all other patents. Therefore, in equilibrium, when all types of consumer goods areproduced, the transfer price of an arbitrary patent must equal V (t).
The cost of inputting Hf (t) units of human capital is w(t)Hf (t), and the revenueof an R&D ﬁrm f is ȧk (t)V (t). Therefore, if the proﬁt of an R&D ﬁrm f , whichinvents the technology for ȧf units of the new product at time t, is denoted by Πf (t),then the following holds:
Πf (t) = ȧf (t)V (t) − w(t)Hf (t),= [λA(t)V (t) − w(t)]Hf (t).

(18)

Because each R&D ﬁrm constitutes only a small portion of the entire R&D sector,
public knowledge capital is considered given for each R&D ﬁrm.
Our interest is in the equilibrium in which product innovation occurs in thesense of ȧf (t) > 0. As Eq. (17) shows, the creation of innovation requires the
input of human capital. Therefore, the condition Hf (t) > 0 must be satisﬁed in an
equilibrium in which innovation occurs. Under the condition Hf (t) > 0, Eq. (18)suggests that, when λA(t)V (t) > w(t), the more resources that are used in R&D,
the greater the proﬁt will be. In this case, ﬁrms have the incentive of trying to
input an unlimited amount of human capital in R&D activities. However, inputof an inﬁnite amount of human capital is not possible in the equilibrium. WhenλA(t)V (t) < w(t), Eq. (18) implies that the input of resources into R&D would
result in negative proﬁt. In this case, the rational decision for the ﬁrm is to not input
resources in R&D, that is, Hf (t) = 0, but this contradicts Hf (t) > 0. Therefore,
in the subjective equilibrium for which free entry to the market is guaranteed andproduct innovation is created, λA(t)V (t) = w(t) must be satisﬁed. Consequently,
the following equation is obtained:
A(t)V (t) =

11

w(t)
.
λ

(19)

2.4

Market Equilibrium

We follow Grossman and Helpman (1991, Ch.3) by assuming that the stock pricesof consumer goods ﬁrms are consistent with V (t), which is the discounted present
value of the proﬁt ﬂow for all ﬁrms producing consumer goods. An individual who
owns stocks from t to t + ∆t receives π(t)∆t in dividends. Also, capital gains ofV̇ (t)∆t can be earned. Therefore, the revenue that a stockholder can earn during
this period is π(t)∆t + V̇ (t)∆t. Incidentally, when funds of V (t) are lent from t
to t + ∆t, the revenue is expressed as r(t)V (t)∆t. In the equilibrium of the asset
market, the revenues of the former and latter must be consistent. Consequently, thefollowing holds:
π(t) + V̇ (t) = r(t)V (t).

(20)

Equation (20) is called the no-arbitrage condition.
We turn now to an equilibrium condition in the consumer goods market. In
the subjective equilibrium of consumer goods ﬁrms, the production amount of all
ﬁrms is x(t). Therefore, the total supply of consumer goods at time t is given by∫ A(t)
x(t)dj = A(t)x(t). On the other hand, the aggregate demand for consumer0
goods is αE(t)/ζ from Eq. (14). Consequently, the following equation holds for theequilibrium in the consumer goods market:
αE(t)= A(t)x(t).
ζ(21)

Market clearing of unskilled labour requires that
ζA(t)x(t) = L(t),where L(t) ≡∫ A(t)0(22)

Lj (t)dj ．The left-hand side of Eq. (22) represents the aggregate

demand for unskilled labour and the right-hand side expresses the aggregate supply.
Finally, market clearing of human capital implies thatȦ(t)= H(t),λA(t)where H(t) ≡∫1

0 Hf (t)df .

(23)

The left-hand side of Eq. (23) represents the aggregate

demand for human capital, while the right-hand side expresses the aggregate supply.

12

2.5

Steady-State Equilibrium

Let us examine the relationship between the ageing of population and long-term
product development performance of the R&D industry. We focus on the steadystate equilibrium of the model and derive the reduced-form equation for the innovation rate. As in Barro and Sala-i-Martin (2004), the steady-state equilibrium is
deﬁned as the situation where all economic variables grow at constant rates in thegeneral equilibrium.
The period of an individual’s education is S(t). Let θ(t) be the ratio of the sumof the number of people receiving education, the number of active skilled workers
and the number of retired skilled workers to the total population. Recall that the
total population at each point in time consists of people receiving education, active
unskilled workers, active skilled workers, and retired people. Then the numberof people receiving education is S(t)θ(t)N/T ; the number of unskilled workers is[1 − θ(t)]N Z/T ; the number of skilled workers is (Z − S)θ(t)N/T ; and the numberof retired people is (T − Z)N/T . The composition of the total population at eachpoint in time is depicted in Figure 2.

N
T
Unskilled workers
θN
T

Retired people
Skilledworkers

O

S

Z

T

q

People receiving education

Figure 2: Composition of the Total Population
Incidentally, the equation below holds for the number of unskilled workers:
L(t) = [1 − θ(t)]

N
Z.
T

(24)

Therefore, the total population N can be expressed asN= S(t)θ(t)N
N
N+ L(t) + [Z − S(t)]θ(t) + (T − Z) .
T
T
T

13

(25)

Suppose now that the human capital per skilled worker, h(t), depends only on thenumber of years in education. We specify the function for human capital formation
as follows:h = eµ(S) ,(26)

where the derivative µ′ (S) represents the gain from the number of additional years
in education, that is, the rate of return on investment in education. We follow Lee(2005) by considering the case in which µ(S) is the linear function µ(S) = µ × S. It
is assumed that µ > 0 is a constant. Consequently, the following equation holds forthe amount of human capital in the macro economy:
H(t) = [Z − S(t)]θ(t)N µS(t)
e
.
T

(27)

Recall that, by deﬁnition, the growth rate of all economic variables is constant in
the steady-state equilibrium. First, Eq. (24) implies that θ(t) and L(t) are constantin the steady-state equilibrium. Also, when θ(t) and L(t) are constant, S(t) mustalso be constant from Eq. (25). Furthermore, using Eqs. (21) and (22), we get
L(t) = αE(t).

(28)

It follows that E(t) is also constant. Therefore, when the economy is in a steadystate equilibrium, the aggregate consumption expenditure of the household sector isconstant at the level E > 0.
We now examine the equilibrium path in which an individual of type i from generation κ maintains a constant consumption level over time. In such a consumption
expenditure path, Ei (κ, s) = Ei (κ, t) holds for all s ̸= t, and so Eq. (9) implies
r(t) = ρ. Dividing both sides of Eq. (20) by V (t) givesπ(t)V̇ (t)+= ρ.
V (t) V (t)(29)

Because V̇ (t)/V (t) and ρ are constant in Eq. (29), the proﬁt rate π(t)/V (t) mustalso be constant. In addition, Eqs. (19) and (22) imply( ζ − α ) λE
π(t)
=
.
V (t)
ζ
w(t)

14

(30)

Equation (30) shows that w(t) is constant. Let g ≡ Ȧ(t)/A(t) be the rate of innovation in a steady state. Considering Eqs. (19), (28), (29), and (30), we get( ζ − α ) λLαζw

= g + ρ.

(31)

Incidentally, an individual has two alternatives: either making an investment in
human capital to work as a skilled worker in the subsequent period or working as
an unskilled worker without receiving education. If an individual of generation κ
engages in labour as an unskilled worker, then the discounted present value of that
person’s lifetime income, Wl (κ), is expressed as∫ κ+Z
Wl (κ) =e−ρ(t−κ) dt

κ

1 − e−ρZ
.
ρ

=

(32)

If an individual of generation κ engages in labour as a skilled worker, then the
discounted present value of that person’s lifetime income, Wh (κ), for S ≤ Z, isgiven by∫ κ+ZWh (κ) ==e−ρ(t−κ) weµS dt

κ+S(e−ρS − e−ρZ )weµS

ρ

.

(33)

An individual wishing to be a skilled worker faces the problem of selecting the
education period with a given time constraint. More speciﬁcally, the individual
solves the following problem:max χ(S) =

1 −ρS
(e
− e−ρZ )weµS ,ρ

s.t. S ≤ Z.
We assume here that limS→0 eρS > eρZ (µ − ρ)/µ. Solving the optimisation problem
yields the following interior solution:
S=

]
1[ (
ρ)
ln 1 −+ ρZ .
ρµ(34)

In the following argument, we deﬁne ν(µ, ρ, Z) ≡ (1/ρ)·[ln(1−ρ/µ)+ρZ] to simplify
the expression.

15

All households are representative agents who have perfect foresight. They compare the discounted present value of their lifetime income that can be earned if they
become skilled workers with that if they become unskilled workers to decide which
type of worker they will become. Such selection can be interpreted as the decision
making for the sector that supplies labour services, which is whether to engage inlabour in the consumer goods sector or in the R&D sector. The subject of ouranalysis is the steady-state equilibrium in which production activities are conductedin both the consumer goods sector and the R&D sector. If a diﬀerence in the discounted present value of lifetime income arises between these sectors, all individuals
should attempt to engage in labour in the sector in which they can earn a higher
discounted present value of lifetime income. Therefore, in the steady-state equilibrium in which production activities are conducted in both the consumer goods sector
and the R&D sector, the discounted present value of lifetime income must be equal
between the sectors. This condition means Wl (κ) = Wh (κ). Using Eqs. (32) and(33), we obtain
1 − e−ρZ(e−ρS − e−ρZ )weµS=.
ρρThis equation can be rewritten as follows:w=

1 − e−ρZ.
[e−ρν(µ,ρ,Z) − e−ρZ ]eµν(µ,ρ,Z)(35)

Therefore, considering Eqs. (24), (31), and (35), we get
ρ+g =( ζ − α ) (1 − θ)λN ZαζTeµν(µ,ρ,Z)[ e−ρν(µ,ρ,Z) − e−ρZ ]1 − e−ρZ.

(36)

Combining Eqs. (23) and (27) with (34) yieldsg=

λ[Z − ν(β, ρ, Z)]θN eµν(µ,ρ,Z) .
T(37)

Consequently, Eqs. (36) and (37) can be used to calculate the values of g and θin the steady-state equilibrium. It is found that g in a steady-state equilibrium isgiven by{}1 ( ζ − α ) λN µν(µ,ρ,Z) [e−ρν(µ,ρ,Z) − e−ρZ ]
g=
Ze
−ρ ,
ϕ
αζ
T
1 − e−ρZ16(38)

where ϕ is deﬁned byϕ≡1+(ζ − α)αζZ[e−ρν(µ,ρ,Z) − e−ρZ ].
(1 − e−ρZ )[Z − ν(µ, ρ, Z)]

In the argument below, we call the rate of innovation in a steady-state equilibrium
the rate of long-term innovation. Equation (38) implies that, as N decreases, gdecreases. Thus, the model predicts that in a knowledge-based economy the rate oflong-term innovation will decline with shrinking population.
For θ in a steady-state equilibrium we get{}ρT
1 ( ζ − α ) µν(µ,ρ,Z) [e−ρν(µ,ρ,Z) − e−ρZ ]
−
,
θ=
Ze
ψ
αζ
1 − e−ρZ
λN(39)

where ψ is deﬁned by
ψ = [Z − ν(β, ρ, Z)]eµν(µ,ρ,Z) +3

Policy Implications

3.1

Ageing and Innovation( ζ − α ) Zeµν(µ,ρ,Z) [e−ρν(µ,ρ,Z) − e−ρZ ]1 − e−ρZαζ.

We ﬁrst examine the inﬂuence of the progress of ageing on innovation and consequent
growth performance. In general, population ageing means that older groups in theage structure of a population increase. The percentage (or proportion) of elderly
people (usually 65 years old and older) in the total population of a country is often
used as a representative indicator of ageing.
In the model, individuals at the age of Z and older are regarded as elderly people,
so the total number of individuals at the age of Z and older (the elderly population)is (T − Z) · (N/T ). Recall that the total population at all points in time is N . If
we express the proportion of the aged population as γ, then we obtain the following
equation:
γ =1−

Z
.
T

(40)

From Eq. (40) the proportion of the aged population, γ, depends on the life expectancy T and the retirement age Z. An increase in T or a decrease in Z dictatesa rise in the proportion of the aged population. Consequently, a comparison of17

the respective steady-state equilibria between those before and after either parameter changes allows an understanding of the eﬀect of the progress of ageing on theendogenous variables.
First, the connection between population ageing and the rate of long-term innovation will be examined. Partially diﬀerentiating Eq. (38) with respect to life
expectancy, we obtain{}∂g
1 ( ζ − α ) λN µν(µ,ρ,Z) [ e−ρν(µ,ρ,Z) − e−ρZ ]
Ze
< 0.
=−
∂T
ϕ
αζ
T2
1 − e−ρZ
In addition, partially diﬀerentiating Eq. (38) with respect to retirement age, we
obtain∂g∂Z=[ e−ρν(µ,ρ,Z) − e−ρZ ]1 ( ζ − α ) λN µν(µ,ρ,Z)e[1 + Z(µ − ρ)]ϕ αζT(1 − e−ρZ )2{}1 ( ζ − α ) λN µν(µ,ρ,Z) [ e−ρν(µ,ρ,Z) − e−ρZ ]
+
Ze
−ρ
ϕ
αζ
T
1 − e−ρZ( ζ − α ) λN[e−ρν(µ,ρ,Z) − e−ρZ ][Z − ν(µ, ρ, Z)](ρZ + e−ρZ − 1)
> 0.
×
αζ
T
(1 − e−ρZ )2 [Z − ν(µ, ρ, Z)]2

Other things being equal, the proportion of the aged population is high foreconomies in which the value of T is high (or the value of Z is low). Consequently,
from the perspective of our model, a decline in the rate of long-term innovation is
expected as the population ages.
Because an increase in Z means a higher mandatory retirement age for workers,
the result, ∂g/∂Z > 0, can be interpreted as an increase in the rate of long-term
innovation along with a rise in the mandatory retirement age. Furthermore, a risein the rate of innovation will contribute to an increase in national income. Thus, themodel implies that encouraging employment of elderly people through an increase in
the mandatory retirement age is important to achieve sustained economic growth.
For many developed countries in which ageing will accelerate in the future, a rise in
the mandatory retirement age will be one focal issue in arguments about employment
policies. The analysis of this study can be considered as a rationale to support thepolicy of encouraging the employment of elderly people, as proposed by Yashiro(1997), OECD (2006), and others.

18

3.2

Ageing and Wage Premium

Next, we consider the direction of changes to the wage premium that accompanies
the progress of ageing. The wage premium is deﬁned as the ratio of the wage earned
by a skilled worker to that of an unskilled worker. The wage rate of unskilled workers
is normalised to 1 and the wage rate of skilled workers is w, and so the wage premium
is w/1 = w. Diﬀerentiating Eq. (35) with respect to retirement age, we get
∂w
∂Z

=

eµν(µ,ρ,Z) [e−ρν(µ,ρ,Z) − e−ρZ ]
−ρZ
)µ].
{
}2 [ρ − (1 − e−ρν(µ,ρ,Z)µν(µ,ρ,Z)ρZ[e− e ]e(41)

From Eq. (41), the sign of ∂w/∂Z is determined by the sign of ρ − (1 − e−ρZ )µ.
We will investigate graphically the relations existing between population ageing andthe wage premium. The curve in Figure 3 is drawn by regarding (1 − e−ρZ )µ as afunction of Z.

µ(1 − e−ρZ )µ∂w∂Z < 0ρ∂w∂Z > 0OeZ

Z

Figure 3: Relationship between Aging and Wage Premium
e then ∂w/∂Z < 0, and if
Figure 3 shows that if the value of Z is less than Ze then ∂w/∂Z > 0. If Z = Z,e then ∂w/∂Z = 0. Therefore,
it is greater than Ze thein an economy in which workers’ mandatory retirement age is higher than Z,
wage premium increases as ageing progresses. In contrast, in an economy in whiche the wage premium declines asworkers’ mandatory retirement age is lower than Z,
ageing progresses. In other words, the eﬀect of population ageing on wage premium
e Thus, our model implies that the eﬀect on the wagechanges at a certain age Z.
disparity between skilled and unskilled workers of promoting the employment of older

19

workers by increasing mandatory retirement age depends on the initial mandatory
retirement age.

3.3

R&D Subsidy

In this subsection we analyse the eﬀect of R&D subsidy policy on industrial innovation. Assume a situation in which R&D subsidy policy is funded with lump sum
taxes paid by households. All revenues from lump sum taxes are allocated to thesubsidies for R&D ﬁrms.
Let us consider the case in which a certain percentage ξ ∈ (0, 1) of the R&D
expenditures of each ﬁrm is aided by the government. The cost of R&D activities
needed by ﬁrm k at time t is w(t)Hf (t). Therefore, this ﬁrm receives a subsidy inthe amount of ξw(t)Hf (t) from the government. Consequently, the total amount of
R&D subsidies from the government at time t is ξw(t)H(t).
We denote the rate of long-term innovation when such R&D subsidies are not
implemented by g ∗ , and the rate of long-term innovation when R&D subsidies are
implemented by g ∗∗ . Therefore, g ∗ is equivalent to the rate of innovation indicatedby Eq. (38). Also, it is easily seen that g ∗∗ is given by{}( 1 − ξ ) ( ζ − α ) λN−ρν(µ,ρ,Z) − e−ρZ ][eg ∗∗ =Zeµν(µ,ρ,Z)
−ρ .
ϕ−ξ
αζ
T
(1 − ξ)(1 − e−ρZ )Comparing of the sizes of g ∗ and g ∗∗ reveals that g ∗∗ > g ∗ holds. Consequently,
R&D subsidy policy increases a ﬁrm’s incentives for R&D investment and helps
achieve an improved rate of innovation.

3.4

National Innovation System

We now review the determinants of innovation, in particular examining the parameter λ on the right-hand side of Eq. (38). Apparently, the greater the value ofλ, the higher the rate of long-term innovation. In Eq. (17), λ was introduced as astructural parameter of the R&D production function model for new product design.
The following relation holds, based on Eq. (17):
ȧf
∂ ȧf
=
= λA.
∂Hf
Hf

20

Therefore, λ is considered to be a factor in the marginal or average productivityof human capital having a new product design as the output. Such a parameter of
productivity can also be interpreted as an indicator of the organisational capability ofan R&D ﬁrm. The organisational capability is a concept in business administrationreferring to the system of organisational routines that facilitates eﬀective productdesign and development.
Organisational capability of ﬁrms depends not only on internal factors, but also
on the institutional foundation of a national economy, which is called the social infrastructure by Hall and Jones (1999). Internal factors include knowledge management such as knowledge cordiﬁcation and application functions within knowledgecreating organisations (Kakabadse, Kouzmin, and Kakabadse, 2001). For the institutional foundation of a national economy, the national innovation system may
provide an important clue. Following Archibugi and Michie (2003), in this article,
the national innovation system is regarded as a nation-speciﬁc factor that has a
crucial role in shaping technical change.
As suggested by the empirical study of Aoki and Branstetter (2010), the importance of having constant access to the outcomes of advanced basic research has been
increasing in the development of industrial technology in recent years. Based on
econometric analyses using ﬁrm-level data, Aoki and Branstetter (2010) conﬁrmed
the signiﬁcance to increased R&D productivity in the United States of increased
knowledge diﬀusion from academic institutions to private-sector industries. The
contribution was evident particularly in areas related to pharmaceuticals, medical
equipment and biotechnology. In comparison to the United States and EU countries,
however, the level of science linkage in the domestic industries of Japan is generallylow.
As for institutions that aim to diﬀuse the outcomes of advanced basic research,
university-industry collaboration constitutes an important element of the nationalinnovation system. Interaction between universities and industries based on ﬂexible system design may facilitate sustained economic growth through active innovation. Accordingly, since 1998, a series of measures to promote university-industry
collaboration has been adopted. At present, Japan’s major universities have estab-21

lished a technology licensing organization (TLO). On the other hand, in a macroeconomic view, the foundation of a TLO in Japan has not necessarily encouraged
university-industry collaboration to function eﬀectively and create innovation. Because university-industry collaboration is positioned, one fundamental point of thenational innovation system, the method of university-industry collaboration in Japan
has considerable room for improvement.
Goto and Woolgar (2006) considered university-industry-government collaboration in the United Kingdom as an example in this regard, and made remarks thatoﬀer various suggestions. Examples include a program called the industrial secondment scheme (ISS) that was adopted by the Royal Academy of Engineering. In thisprogram, a university lecturer works in industry for 3-6 months and acquires practical experience in the applicability of their own research. Part or all the lecturer’ssalary during this period is paid by the Royal Academy. Another example, for transferring technology from a university to industry through participation by graduate
students, is the knowledge transfer partnership (KTP) system. In this program,
graduate students participate in research and development projects run jointly by
the university and companies as KTP Associates, and enhance the research level in
the area of science and technology relevant to industry by considering the directionof research demanded by the industrial world. Through human resource development in the KTP program, industries can expect to employ researchers who are
equipped with advanced research knowledge from universities and industrial R&D
expertise from their practical experience.
In a knowledge-based economy accompanied by population decrease and aging,acquiring researchers in science and technology will be increasingly important forindustries. In the past, Japanese companies have not been active in hiring graduates
who have a doctoral degree. However, the increase in the number of students who
have acquired explicit and tacit knowledge of industrial R&D, through programs
such as KTP described above, and skills in advanced research is expected to be
beneﬁcial for both universities and industries. Programs in the manner of ISS andKTP are yet to be implemented in Japan, and are therefore expected to be rather
informative for innovation policy makers in Japan.

22Movement of people between industry and universities has been institutionallydiﬃcult in Japan, which has limited the opportunities for a close exchange of knowledge as a result. A system of interaction between industry and academia basedon ﬂexible system design would facilitate sustainable growth through active innovation. Nonetheless, research into the national innovation system from the perspectiveof social science has not been adequate. Development of eﬀective policy prescriptions requires multidisciplinary knowledge encompassing economics, management
engineering, technology management, and other ﬁelds.

4

Conclusion

The underlying mechanism behind long-term growth performance associated withpopulation ageing is yet to be adequately understood. However, very few studies in
the literature have attempted to examine links between population ageing and innovation. This article is an attempt to ﬁll that void. Speciﬁcally, a main contributionof this article is to present a manageable framework for analysing the macroeconomic
eﬀects of population ageing, extending the model of Grossman and Helpman (1991,
Ch.5).
We show that the rate of technical progress will decline with population ageing, because population ageing inhibits technical progress and consequently also
economic growth. In addition, we show that raising the mandatory retirement age
aids technical progress. This suggests that encouraging the employment of elderly
people is important for promoting economic growth. On the other hand, raising the
mandatory retirement age may cause greater wage income disparities depending on
the rate of return on investment in education and the rate of time preference in theeconomy.
How does raising the mandatory retirement age aﬀect growth performance?
Based on our model, the theoretical mechanism might be explained intuitively as
follows. First, a higher retirement age increases the number of active skilled workers
engaging in R&D activities. This has a direct positive eﬀect on innovation. Second, an increase in the retirement age also increases the number of active unskilledworkers. As a result, proﬁt from the production of consumer goods increases, which23generates increased R&D revenue to promote product innovation. Third, raising
the retirement age means a longer period of time during which skilled workers areable to earn income. Thus, the incentives for economic agents to gain skills througheducation and training will increase. As a result, there is an increased accumulationof human capital in the macro-economy that will increase the rate of innovation and,
therefore, national income.
Despite our simplifying assumptions, our model captures salient features of thereal world. We hope that it will serve as a stepping-stone towards developing future
work.

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26