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Demand Creation and Economic Growth

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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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Demand Creation and Economic Growth
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.

Demand Creation and Economic Growth
ABSTRACT
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.
JEL Nos. E1, E2
Key words: Endogenous Growth, Demand Creation, Endogenous Preferences.
1

Demand Creation and Economic Growth
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

I. The Model
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.
A. Final Goods
Final goods are produced with an intermediate good as the only input. Production of all the final
goods requires the same intermediate good X. Production function is also common.
(1)
y = AX
(0 < A < 1)
k
k
We assume the perfect competition. Therefore, zero profits ensue:
(2)
P (t ) y (t) = P (t )X (t)
k
k
X
k
Here P (t) is the price of the k-th final product, and P (t ) the price of intermediate good.
k
X
Because of the common linear production function (1), the zero profit condition (2) is equivalent to
(3)
P (t )A = P (t)
k
X
Thus, we can adjust unit of final products in such a way to make all the prices of final goods one.
Then
P = A < 1
X
Output of each final good is equal to its demand D (t) no matter how the latter is
k
determined.
(4)
y (t) = D (t)
k
k
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 D (t) follows the logistic curve:
k
D
µ
(5)
D t
( ) = [
0
? t
D
? 0 + (µ ? D )e µ
? 0
]
Since the mechanism is the same for all the products or sectors, for the moment, we drop
k for
clearness and write D (t) as D(t) . We will explore microeconomic foundations for the logistic
k
growth of demand in section III; In section I and II, (5) is taken as given. D in (5) is the initial value
0
of D(t) . Starting with D smaller than µ / ?, D(t) initially increases almost exponentially, but its
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 u (u>0), we can write D t
( ) = un for integer n . D(t)
t
t
instantaneously changes by either u or - u. And assume that the probability that D(t) increases from
nu to (n+1)u between t and t + ?t is µn t
? + o( t
? ) while the probability that D(t) decreases from
nu to (n- 1)u is ?
n2?t + o(?t ) with both µ and d positive. This is a typical birth and death process.
Note that the ‘birth rate’ µ is constant while the ‘death rate’ ?
n is increasing in n. The idea is that
when a product or an industry gets older and n becomes larger, the probability that the product is
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 D(t) = un as P(n, t). Then under the above assumptions,
P(n, t ) satisfies the following differential equation:
dP(n, t )
(6)
= µ(n ? )
1 P(n ? ,
1 t) +
(
?n + )
1 2 P(n + ,
1 t ) ? (µ + n
?)nP(n,t)
dt
We denote the expected value (the first moment) of n(t ) or equivalently that of D(t) by
D(
ˆ t) ; i.e.
3

D(
ˆ t) = ?? nP(n,t)
n=1
Then given (6), D(
ˆ t) satisfies the following differential equation1:
dD(
ˆ t)
dP(n, t)
=
n
??
(7)
dt
n=
dt
1
2
= D
µ (
ˆ t ? D
? (
ˆ
)
t)
Equation (7) can be solved to obtain the logistic equation (5); D(t) in (5) is to be interpreted as the
expected value D(
ˆ t) of D(t) in this birth and death process.
To obtain the logistic equation for D(
ˆ t) , we assumed above that the ‘birth rate’ µ was
constant while the ‘death rate’ ?
n was increasing in n . Alternatively we can assume that the birth
t
t
rate µ / n is decreasing in n while the death rate d is constant. Diffusion of a product may lead to
t
t
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 / n and d, D(
ˆ t) satisfies
t
the following equation:
D
d (
ˆ t) = µ ? D
? (
ˆ t)
dt
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 individual product or industry grows exponentially. Rather demand for or
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 y (t) follows the logistic growth of demand; y (t) also
k
k
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
N as if it were constant.
B. Intermediate Good
To keep our model as simple as possible, we assume that there is only one kind of
intermediate good X, and that X is produced by using capital K alone:
(8)
X = K.
Here X is the sum of intermediate goods used in production of final goods:

1 To be precise, D(
ˆ t) satisfies
dD(
ˆ t)
dp(n, t)
=
n
??
dt
n=
dt
1
= D
µ (
ˆ t ? D
? (
ˆ
)
t)2 ? V
? (t).
V (t ) is the variance of n(t ) or D(t) with the differential equation defined in terms of D(
ˆ t) , V (t)
and K (t ) where K (t) is the third central moment or kurtosis of n(t ) or D(t) . It can be shown that
3
3
with D(
ˆ ? ) = µ /?, both V (? ) and K (? ) are zero.
3
4

N
X =
X
? k
k =1
We note that production function (8) has unitary elasticity of capital, and therefore that as
long as capital accumulates, X grows without limit. And given the common function of final good (1),
whenever X grows, production of final good can also grow. However,
X is intermediate good, and as
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)
P X (t ) = P K(t)
X
X
Investment, namely an increase in K on the other hand requires finished goods as an input.
To be specific, we assume that an increase in K at the rate of z requires ? (z) of final products. ? (z)
satisfies
(10)
? ?
( z) > ,
0
? ?(z) > 0 for z = K& K >0 with ?(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) S is then given by
?
?
(11)
S = ? [P K ?( )
exp
?
?
? ?
z K
?
? ]
{? ? ut }
du d
t
X
t
where ? (z)K is investment. We note that S satisfies
t
&
?
S
(P
( z ))K
t
X
? t
t
=
+
?
t
S
S
t
t
and observe, therefore, that ? is the rate of return on ‘stock’ of this firm or the interest rate. Given
t
the initial capital stock K , the firm maximizes (11) under the assumptions in (10). Since ? (z) is
0
convex, the optimum capital accumulation z is uniquely determined when it exists (Uzawa [1969]).
t
By means of the maximum principle, we know that it satisfies
?
(12)
?
( z ) = ??
?
[P ? ?(z? )]exp
t
{? (? ?
?
z )du
t
}d
t
X
u
u
?
The right-hand side of (12) can be interpreted as Tobin’s marginal q; For optimality of the firm’s
investment decisions, the marginal cost of investment must be equal to the marginal q. With
production function (8), it is plain that growth of intermediate goods also satisfies (12). Given (1), so
does the growth of final goods.
C. Emergence of New Final Goods or Industries
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 Basic Science and Technology Statistics, the share of public money in total R & D
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 t and
t + ?t is ?N?t where N is the number of existing final goods (?>0). Since an invention or an
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 N; The more the number of products or
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 t, N(t) is equal
to N, Q(N, t) satisfies the following equation.
dQ
(13)
= ? NQ
? (N,t) + ?(N ? )
1 Q( N ? ,
1 t )
dt
The appendix shows that the solution of this equation under the initial condition
Q(N ,0) = (
?N ? N ) = (
?N ? )
1
0
is
? ?
? ?
?
(14)
Q N t = e t 1 ? e t N
( , )
(
) 1
The probability that there are N goods at time t and the N + 1-th good emerges during t and
t + ?t is then given by
?
?
t
? ? N 1
(15)
?NQ(N,t)?t = ?
?
Ne
(1? e t) t?
At time t, the production of final good which emerged at ? ?
( < t) y? (t) has grown to
µ
(16)
y (t =
?
)
?+ (µ ?
e? µ t? ?
?) ( )
since the growth of y t
? ( ) obeys the logistic curve. Without loss of generality we can assume that the
initial production of newly invented good D to be 1 in equation (5). This is the structure of the
0
economy.
II. Growth of the Macroeconomy
In this section, we will analyze the growth of the macroeconomy in this model.
6

A. The Basic Result
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 Y(t). Y(t) is simply the sum
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, P X (t ) which is equal to ? AX
y .
k =?
X
k
k
k
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
?
t
??
??
1
µ
Y(t ) =
Ne?
?
(
?
1? e?
N
) ? y (t )d
?
? +
?0
(
? µ
=1
+
? (µ ? )e t
?
)
N
(17)
?
t
=
??
??
1
µ
Ne?
?
µ
(
?
1? e?
N
) ?
?0
1
[
d? +
+
µ
?
? µ
=
? (µ ? )
?e? (t? )]
( +
? (µ ?
e t
)
?
)
N
The second term of the right hand side is simply output of the
‘first’ sector at time t, y (t ) . Using
0
? ? ??
? ?? N ? 1
d
? ??
Ne
(1 ? e
)
=
(1 ? e
N
)
d?
and
?
(1
? ? ?e?? )N = e?? ? 1
N 1
=
we obtain
? d
??
?
(e
? )
1 µ
t
d
?
? ?
?
?
µ
Y(t ) = ?0[
d? +
+
µ
?
?
?
µ
(µ ?
)
?e? (t? )]
( +
? (µ ?
e t
)
?
)
??
t
e
µ
(18)
= ?
?
µ
0 [
d? +
+
µ
?
?
?
µ
(µ ?
e? (t
)
? )
?
] ( +? (µ? e t
)
?
)
t
e (
? t? u)
= ?
µ
µ
?0[
du +
+
µ
?
?
µ
(µ ?
e? u
)
?
] ( +? (µ? )e t
?
)
From (18), the growth rate of GDP becomes
Y&(t )
f (t )?? f&(t )?
g =
= ?+ ?
?
?
t
Y(t )
?
?Y(t)??? f (t)?
where f( t) is the logistic equation:
µ
f (t ) = ( +
? (µ ?
e? t
)
? µ )
It is easy to show that g satisfies
t
? µ
(19)
&
g = ( g ?
)[
? (
2 µ ?
)e t
?
f (t) ? µ ? g ]
t
t
t
with initial value g .
0
Y&(t )
g =
? µ ?
= =
+
?
0
Y (t t 0
)
? µ
Also, since e t f (t ) approaches zero, we can establish that the growth rate of GDP asymptotically
approaches ?.
&
Y(t )
lim g =
= ?
t
t? ?
Y(t )
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
t
growth rate defined as
g / t
? ? 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 g satisfies the following equation:
Y&t
g =
= n 1
( ? ? )[log *
Y ? logY ]
t
t
Yt
Here
*
Y is the steady state level of per-capita Y, n, the growth rate of population, and a is the capital
elasticity. Starting with the initial value Y below
*
Y , the growth rate decelerates over time. The
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.
B. An Extension: The Non-Poisson ‘Polya urn’ model
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 , p? is
?
p =
?
? + ? (? > 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
p? rather than as a branch off from the existing goods, namely that p? is independent of the number
of existing goods. In this case, when we denote the probability that there are
N goods at t by Q(N ,? )
as we did it previously, then Q(N ,? ) satisfies
Q(N ,? + )
1 = (1 ? p Q
) ( N ,
+
?
?
?) p Q(N
,
?
1 ? )
? ?
=
?
Q(N ,?
?
)
Q(N
,
1 ? )
?
?
?
?
+ ???
+ ?
?
?
?
+ ???
?
for t =1, 2, …
with the following boundary conditions:
? 1
? 2
? ? ? 1
Q( ,
1 ? ) =
L
?
?
?
?
+ 1?? ?
?
?
?
+ 2??
?
?
?
?
+ ? ? 1??
and
?
?
?
?
Q ?
( ,? ) =
=
? ?
(
+
?
)(
1
+ 2)L ?
(
+ ? ? )
1
[?]?
?
where [? ] is defined by the equation.
The solution of this equation is
c ?
( , k ? k
)
Q(k,? ) =
[]?
?
where c(t , k) is the absolute value of the Sterling number of the first kind: See Aoki [1997, p. 279
] or
Abramovitz/Stegun [1968, P. 825]. Using the generating function
[
k
k
x] =
c k j x j
? ( , )
j=0
we obtain the expected value of GDP, Y(t) as
9

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