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In the standard literature, the fundamental factor to restrain economic growth is diminishing
returns to capital. This paper presents a model in which the factor to restrain growth is saturation of
demand. We begin with common observation that growth of an individual product or sector grows fast
at first, but its growth eventually declines to zero. The economy sustains growth by the introduction
of new products/industries. Preferences are endogenous in this model. The introduction of new
products/industries affects preferences, and creates demand. By so doing, it induces capital
accumulation, and ultimately sustains economic growth.

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Masanao Aoki and Hiroshi Yoshikawa*

February 1999

* Department of Economics U. C. L. A. and Faculty of Economics, University of Tokyo, respectively. We

thank seminar participants at Tohoku, Osaka, Keio, Siena and Tokyo Universities. We want to give special

thanks to Professors Kazuya Kamiya, Yoshiyasu Ono, Etsuro Shioji and Robert M. Solow who read the

paper very carefully and provided us with very useful comments and suggestions. All remaining errors are,

of course, ours.

In the standard literature, the fundamental factor to restrain economic growth is diminishing

returns to capital. This paper presents a model in which the factor to restrain growth is saturation of

demand. We begin with common observation that growth of an individual product or sector grows fast

at first, but its growth eventually declines to zero. The economy sustains growth by the introduction

of new products/industries. Preferences are endogenous in this model. The introduction of new

products/industries affects preferences, and creates demand. By so doing, it induces capital

accumulation, and ultimately sustains economic growth.

Key words: Endogenous Growth, Demand Creation, Endogenous Preferences.

1

In the standard literature, the fundamental factor to restrain economic growth is diminishing

returns to capital. In this paper, by presenting a model we suggest that ‘saturation of demand’ is

another important factor to restrain growth.

In the less mathematical literature and casual discussions, the idea of ‘demand saturation’ has

been very popular. In fact, plot a time series of production of any representative product such as steel and

automobile, or production in any industry against year, and with few exceptions, one obtains a S-shaped

curve. Figure 1 due to Rostow [1978] demonstrates this

‘stylized fact’. The experiences of diffusion of

consumer durables such as refrigerator, television set, car, and personal computer tell us that deceleration of

growth comes mainly from saturation of demand rather than diminishing returns in technology. Growth of

production of a commodity or in an individual industry is bound to slow down because demand grows fast

at the early stage but eventually its growth necessarily slows down. Thus the demand for some products

grows much more rapidly than the GDP, while the demand for others grows much more slowly.

Products/industries face different income elasticities of demand. The celebrated Engel Law based on

saturation of demand for food is merely an example.

Unfortunately, the existing literature on growth abstracts largely from this important fact that

products/industries face different income elasticities of demand and that each product/

industry

experiences a typical S-shaped life cycle; The standard assumption in the literature is that all the

products are ‘symmetric’ and income elasticities of demand are common for all the products, namely

that the elasticities are one for all. This standard assumption simply contradicts common observations

such as the Engel Law and Figure 1. In place of it, this paper takes the logistic growth of an individual

product/industry as a ‘stylized fact’, and presents a formal model of growth built on this ‘stylized fact’.

An obvious implication of the logistic growth of an individual product/industry is that the

economy enjoys high growth if it successfully keeps introducing new products or industries for which

demand grows fast. Despite its popularity, however, the idea of demand saturation/creation has never

been formally analyzed presumably because of apparent mathematical difficulty. In this paper, we

postulate that growth of each commodity or sector decelerates over time and goes asymptotically

down to zero. The ‘age’ of a product or sector is essential in our analysis. Innovations based on

learning by doing in production bear new commodities or sectors which enjoy high growth of demand,

and by so doing sustain economic growth of the economy as a whole.

We do not mean to suggest that our model is a substitute for the existing literature, but

believe that it sheds lights on a neglected and yet very important aspect of innovations and economic

growth. The existing literature focuses on the issue of whether the elasticity of capital in production

function is one (the so-called AK model) or less than one (the ‘old’ Solow model). It concerns the

property of production function or technology. We maintain that not only diminishing returns on

capital in production but also saturation of demand for existing products/sectors is a very important

factor to limit economic growth. We also maintain that in addition to the standard total factor

productivity (TFP) growth, namely an ‘upward shift’ of production function, technical progress

creates demand.

To substantiate this argument, in section I, we present a model which incorporates the basic

idea. We begin with demand for individual product rather than preferences since the former is more

directly related to the stylized fact than the latter. Section II studies growth of the economy as a

whole. Out of steady state, ‘vigor of demand’ determines growth while the ultimate factor to sustain

economic growth in steady state is creation of new products/industries. Under the standard Poisson

assumption, successive creation of new products/industries sustains steady state growth. However, we

demonstrate that under the alternative ‘Polya urn’ assumption that the success probability of

innovation gets smaller as time goes by, the growth rate of the economy must decelerate and go

asymptotically down to zero. Section III provides microeconomic foundations for demand. By so

doing it suggests two different macroeconomic models, one the Ramsey model with the representative

consumer, and the other with diffusion of goods among different households. Two models suggest

different interpretations of saturation of demand. Finally, section IV offers some concluding remarks.

2

We study an economy in which heterogeneous final goods and an intermediate good are

produced. In this section, we take demand for each final product as given, and concentrate on

production. Section III considers the consumer

’s behavior and provides microeconomic foundations

for demand. Let us begin with final goods.

goods requires the same intermediate good

(1)

(0 <

We assume the perfect competition. Therefore, zero profits ensue:

(2)

Here

Because of the common linear production function (1), the zero profit condition (2) is equivalent to

(3)

Thus, we can adjust unit of final products in such a way to make all the prices of final goods one.

Then

Output of each final good is equal to its demand

determined.

(4)

In this section, we take a S-shaped life cycle of demand for each product/industry as a stylized fact.

To make our analysis tractable, assume that

µ

(5)

( ) = [

0

?

? 0 + (µ ?

? 0

]

Since the mechanism is the same for all the products or sectors, for the moment, we drop

clearness and write

growth of demand in section III; In section I and II, (5) is taken as given.

0

of

0

growth eventually decelerates, and approaching its ‘ceiling’ µ / ?, the growth rate declines

asymptotically to zero. A typical shape of the logistic growth is illustrated and compared with

exponential growth in Figure 2.

There are several ways to interpret the logistic growth. One way is to interpret it as the

expected value of a particular stochastic process. Consider, for example, the following birth and death

process. With an appropriate unit

( ) =

instantaneously changes by either

? +

? ) while the probability that

Note that the ‘birth rate’ µ

when a product or an industry gets older and

replaced by other new products or the industry falls behind new industries becomes higher. We shortly

show that an alternative specification is possible.

We write the probability that

(6)

= µ(

1

1

(

?

1 2

1

?)

We denote the expected value (the first moment) of

ˆ

3

ˆ

Then given (6),

ˆ

ˆ

=

??

(7)

1

2

=

µ (

ˆ

? (

ˆ

)

Equation (7) can be solved to obtain the logistic equation (5);

expected value

ˆ

To obtain the logistic equation for

ˆ

constant while the ‘death rate’ ?

rate µ /

eventual deceleration of growth of demand as is often the case for consumer durables, and/or room for

quality improvement of a product which presumably raises demand for it may get narrower as the

product becomes older. In any case, under this alternative assumption of µ /

ˆ

the following equation:

ˆ

? (

ˆ

instead of (7). Since the qualitative results obtained are basically the same as for the logistic case, in

what follows we will keep the logistic assumption. We simply note that the diminishing ‘birth rate’ of

demand can be handled in a similar way.

Though exponential growth is often taken for granted by economists, there is actually ample

evidence to show that no

production of a product, or an industry typically grows drawing the logistic curve. In fact, a

n eminent

mathematician Montroll [1978] goes so far as to suggest that almost all the social phenomena, except in

their relatively brief abnormal times, obey the logistic growth. Figure 1 well demonstrates this well-

known fact of life in our economy.

Production of each final product

satisfies equation (5). So far, we have focused on a final good. The number of final products is not

given, however. Rather at every moment a new product or sector emerges. The emergence of an

utterly new final good or a new sector is the result of innovations. Before we explain it, we turn to

production of intermediate goods taking the number of final goods

To keep our model as simple as possible, we assume that there is only one kind of

intermediate good

(8)

Here

1 To be precise,

ˆ

ˆ

=

??

1

=

µ (

ˆ

? (

ˆ

)

? (

ˆ

and

3

3

with

ˆ ? ) = µ /?, both

3

4

?

We note that production function (8) has unitary elasticity of capital, and therefore that as

long as capital accumulates,

whenever

we have seen it previously, growth of demand for each final good decelerates and declines eventually

to zero. In this model, the factor to limit growth is not diminishing returns on capital but declining

growth of demand.

Capital accumulates so as to maximize the value of this industry (firm). Profit of this

industry stems from selling intermediate goods to firms producing final goods, and is

(9)

Investment, namely an increase in

To be specific, we assume that an increase in

satisfies

(10)

? ?

(

0

? ?(

0

? (0

? ) = 1.

For simplicity, we assume that capital does not depreciate. Assumption (10) means the standard

convex adjustment cost of investment. Romer [1986], among others, makes this assumption.

The value of this industry (or firm)

?

?

(11)

exp

?

?

? ?

?

? ]

{? ?

where ? (

&

?

(

(

?

=

+

?

and observe, therefore, that ? is the rate of return on ‘stock’ of this firm or the interest rate. Given

the initial capital stock

0

convex, the optimum capital accumulation

By means of the maximum principle, we know that it satisfies

?

(12)

?

(

?

[

{? (? ?

?

}

?

The right-hand side of (12) can be interpreted as Tobin’s marginal

investment decisions, the marginal cost of investment must be equal to the marginal

production function (8), it is plain that growth of intermediate goods also satisfies (12). Given (1), so

does the growth of final goods.

So far we have taken the number of final goods as if it were constant. In fact, new final

goods and/or industries always emerge as a result of innovations. We can flexibly interpret final

‘goods’ as ‘sectors’ or ‘industries’ if we wish.

Much effort has been made to explicitly analyze R and D activities and inventions in growth

models. In fact, the achievement of the ‘endogenous growth theory’ is to have combined growth

models with models of R & D activities. Certainly, in the advanced economies a substantial part of R

and D is done by private firms from profit motives. It is not the whole story, however, because R and

D of non-profit motives also plays an important role. Research activities at universities is an obvious

example. In Japan (1996), for example, out of 15 trillion yen annual national R & D expenditures, 3

trillion yen finances researches at colleges and universities, 2 trillion yen goes to public research

institutions, and the remaining 10 trillion yen is spent by private firms for their R & D activities.

Roughly one third of the broadly defined R & D activities are done by non-profit motivated

institutions. In many countries, government also spends considerable money on R & D. According to

the OECD

expenditures is 33.6% for U. S. (1995), 22.9% for Japan (1995), 37.1% for Germany (1991), and

44.3% for France (1993). We also note that borrowed technology plays an important role for late

comers, and that borrowed technology is not necessarily fully patented.

5

In any case, profit-motivated R & D activities have been already well analyzed in the

existing literature. Grossman and Helpman [1991], for example, present ‘product variety’ and ‘quality

ladder’ models. In the former, profit-motivated R & D activities keep generating new products, and an

expanding product variety, by way of the so-called Dixit/Stiglitz utility/production function, brings

about ever higher level of utility or production. In the latter, given the number of commodities, R & D

investment improves quality of the commodities. Aghion and Howitt [1992], on the other hand,

present a model in which successful innovations raise economywide efficiency.

These works no doubt shed much light on important aspects of innovations and economic

growth. However, given the complexity of the way in which technical progress affects economic

growth, much remains to be done. The existing literature based on the

Dixit/Stiglitz production/utility

function or otherwise, in effect, endogenously explains total factor productivity. In what follows, we

make an attempt to formalize the hitherto neglected aspect of technical progress, namely ‘demand

creation’ due to technical progress, and study how it affects economic growth. Our primary interest is

not in microeconomic foundations for R & D activities but in the way in which technical progress

affects the economy. Not to minimize the importance of profit motives of private firms to do R and D

but to simplify the analysis for our purpose to focus on a different problem, the present analysis

following Arrow [1962] and Stokey [1988] abstracts from profit maximization.

We assume that an invention of a new final good or an emergence of a new sector stems

stochastically from learning in the process of production of the existing products. To be specific, we

assume that the probability that a new final good is invented or a new industry emerges between

emergence of new sector is a branch off from an existing good or sector, the rate of success probability

is proportional to the number of existing final goods/sector

sectors in the economy, the more likely a new product or sector emerges.

? is a parameter to represent

the strength of innovations or more precisely the probability that a new good or industry emerges in

the existing process of production. Innovations are thus accidental, but the prior

‘knowledge’ and

experiences which stem from the existing production is essential to them. In this respect, we follow

Arrow [1969] who argues that “the set of opportunities for innovation at any one moment are

determined by what the physical laws of the world really are and how much has already been learned

and is therefore accidental from the viewpoint of economics”.

Given this assumption, the probability that the number of final goods at time

to

(13)

= ?

? (

1

1

The appendix shows that the solution of this equation under the initial condition

?

?

1

0

is

? ?

? ?

?

(14)

( , )

(

) 1

The probability that there are

?

?

? ?

(15)

?

?

(1?

At time

( <

µ

(16)

?

)

?+ (µ ?

?) ( )

since the growth of

? ( ) obeys the logistic curve. Without loss of generality we can assume that the

initial production of newly invented good

0

economy.

In this section, we will analyze the growth of the macroeconomy in this model.

6

The aggregate value added or Gross Domestic Product (GDP) of this economy is stochastic,

but in what follows, we will focus on its expected value and denote it by

of production of all the final goods. Since profits in the final good sectors are zero by the assumption

of perfect competition, the aggregate value added is equal to the value added (profit) of the

intermediate good sector,

Figure 3 illustrates this model economy. Each sector once it emerged grows logistically.

New sectors emerge stochastically, and the aggregate value added or GDP is simply the sum of outputs

of all the then existing sectors.

From (15) and (16), we know that the expected value of GDP of this economy is given by

?

??

??

1

µ

?

(

?

1?

) ?

?

? +

?0

(

? µ

=1

+

? (µ ? )

?

)

(17)

?

=

??

??

1

µ

?

µ

(

?

1?

) ?

?0

1

[

+

µ

?

? µ

=

? (µ ? )

?

( +

? (µ ?

)

?

)

The second term of the right hand side is simply output of the

‘first’ sector at time

0

? ? ??

? ??

? ??

(1 ?

)

=

(1 ?

)

and

?

(1

? ? ?

=

we obtain

?

??

?

(

? )

1 µ

?

? ?

?

?

µ

+

µ

?

?

?

µ

(µ ?

)

?

( +

? (µ ?

)

?

)

??

µ

(18)

= ?

?

µ

0 [

+

µ

?

?

?

µ

(µ ?

)

? )

?

] ( +? (µ?

)

?

)

?

= ?

µ

µ

?0[

+

µ

?

?

µ

(µ ?

)

?

] ( +? (µ? )

?

)

From (18), the growth rate of GDP becomes

= ?+ ?

?

?

?

?

where

µ

? (µ ?

)

? µ )

It is easy to show that

? µ

(19)

&

)[

? (

2 µ ?

)

?

with initial value

0

? µ ?

= =

+

?

0

)

? µ

Also, since

approaches ?.

&

lim

= ?

7

The growth rate of the economy is initially higher than? by µ- d, but it eventually goes down to ?.

The exact time path depends, of course, on µ, d, and ?.

It is important to recognize that not only the steady state growth but also the out of steady

state growth is generated by the successive emergence of new products/industries. The growth of

older industries keeps declining while newer products/industries enjoy high growth. How high

depends on µ and d. From the perspective of this model, it is easy to understand that historians have

identified the ‘leading’ or ‘key’ industries in the process of economic growth. The best known

example would be perhaps Rostow [1960, pp.261-62] who argues that

“The most cursory examination of the growth patterns of different economies, viewed

against a background of general historical information, reveals two simple facts:

1. Growth-rates in the various sectors of the economy differ widely over any given period of

time;

2. In some meaningful sense, over-all growth appears to be based, at certain periods, on the

direct and indirect consequence of extremely rapid growth in certain particular key

sectors.”

Vigor of the leading sectors depends on µ and d in the model. For the sake of illustration, we

show a simulation result (Table 1 and Figure 4). In this example, we assume that ?, µ, and d are 0.03,

0.12 and 0.02, respectively. Table 1 and Figure 4 show both the growth rate of GDP and the average

growth rate defined as

? ? for each period (year). For the first ten years, the growth rate of the

?=1

economy is higher than 9%. In the year 20, it is still 5.7%. It is the year 40 when the growth rate

slows down to 3.2% which is close to the assumed asymptotic rate 3%. The average growth rate, of

course, decelerates much more slowly than the growth rate itself. The average growth rate for the first

thirty years, for example, is 7.5% although the growth rate in the year 30 is 3.9%. This example

demonstrates that depending on µ andd, the economy can sustain a much higher growth rate than the

equilibrium rate for a very long period. To repeat, the deceleration of growth comes not from

diminishing returns to capital but from saturation of demand.

Everyone knows that no economy grows at 10% indefinitely. Some economies, however,

actually experienced the 10% growth for a decade, and this decade long high growth is often crucial

for their growth experiences. Japan, for example, kept the 10% growth for a decade and a half from

1955 through 1970. As of 1955, almost a half of working population in Japan was in agriculture. The

era of high economic growth had transformed a semi-traditional economy into a modern industrial

nation. We cannot dismiss ‘out of steady state’ merely as transitory, but must attach equal importance

to it as to the steady state.

The out of steady state growth path illustrated in Figure 4 is qualitatively similar to that

obtained in the old Solow [1956] model; Namely the growth rate decelerates over time. The

mechanism is different, however. In the Solow model, diminishing returns to capital in production is

the factor to bring about slower growth. To be specific,

Mankiw, Romer and Weil [1992] show that in

the standard Solow model, the growth rate

=

( ? ? )[log *

Here

*

elasticity. Starting with the initial value

*

0

extent of deceleration depends on the capital elasticity a.

In contrast, in the present model the deceleration of the ou

t of steady state growth rate comes

from saturation of demand. To be specific, as equation (17) shows, the out of steady state growth path

depends on µ and d which determine how soon demand reaches its saturation.

In the model above, we assumed that the instantaneous probability that a new good (or

sector) emerged in the process of production of each existing good obeyed the Poisson distribution

with the parameter ?. This is the standard assumption in the literature. However, it is interesting to

8

explore what happens for the ultimate growth when the probability gets smaller and smaller as time

goes on. This question is important when opportunities for innovations diminish as time goes by. It is

in fact often suggested that growth of ‘mature’ or ‘old’ economy slows down because such

opportunities diminish. Kuznets [1953], for example, argues that

“In the industrialized countries of the world, the cumulative effect of technical progress in a

number of important industries has brought about a situation where further progress of

similar scope cannot be reasonably expected. The industries that have matured

technologically account for a progressively increasing ratio of the total production of the

economy. Their maturity does imply that economic effects of further improvements will

necessarily be more limited than in the past.”

Based on the American experiences, McLaughlin and Watkins [1939] share this kind of pessimism.

Since the Poisson distribution is so commonly assumed for a success in R & D in the

existing literature (e.g. Grossman and Helpman [1991] and Aghion and Howitt [1992]), it is

interesting to check what happens for economic growth when the assumption does not hold. To

answer this question, we analyze a discrete-time model.

In place of the Poisson distribution, we assume that the probability that a new good or sector

emerges at t ,

?

?

? + ? (? > 0, t=1, 2, … )

This probability decreases in t , and declines asymptotically to zero. This kind of model often called

‘Polya-like urns’ is extensively used in population genetics; See, for example, Hoppe [1984].

To simplify our presentation, we assume first that a new good is invented

‘exogenously’ with

of existing goods. In this case, when we denote the probability that there are

as we did it previously, then

1 = (1 ?

) (

+

?

?

?)

,

?

1 ? )

? ?

=

?

?

)

,

1 ? )

?

?

?

?

+ ???

+ ?

?

?

?

+ ???

?

for t =1, 2, …

with the following boundary conditions:

? 1

? 2

? ? ? 1

1 ? ) =

L

?

?

?

?

+ 1?? ?

?

?

?

+ 2??

?

?

?

?

+ ? ? 1??

and

?

?

?

?

( ,? ) =

=

? ?

(

+

?

)(

1

+ 2)L ?

(

+ ? ? )

1

[?]?

?

where [? ] is defined by the equation.

The solution of this equation is

( ,

)

[]?

?

where

] or

Abramovitz/Stegun [1968, P. 825]. Using the generating function

[

? ( , )

we obtain the expected value of GDP,

9

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