Discussion Paper No. 510

EMERGENCE OF NEW INDUSTRIESAND ENDOGENOUS GROWTH CYCLES

Ryo Horii

June 2000

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Emergence of New Industries andEndogenous Growth Cycles

Abstract
This paper constructs a growth model in which monopolistically competing firms choose the characteristic of their own product from an unbounded
product space. While consumers wish to satisfy various needs by purchasing
a diverse range of goods, production costs are lower for those goods that are
more similar to existing ones because of spillover eﬀects in the learning-bydoing process. The dynamic interaction of these centrifugal and centripetal
forces on the product space leads to sporadic emergence and disappearance
of industries, which is a source of cyclical fluctuations in the macroeconomic
growth rate.
JEL Classification Numbers: O41, E32, L16.
Keywords: product diﬀerentiation, learning-by-doing with spillover, emergence of industries, growth cycles.

Ryo Horii
The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki
Osaka 567-0047 JAPANE-mail address: horii@iser.osaka-u.ac.jp

1

Introduction

Beginning with the seminal works of Romer (1987, 1990) and Grossman andHelpman (1989), many models have been presented in which an increase in
the number of goods drives economic growth. Most of these models predictthat the economy will eventually converge to a balanced growth path, where
the economy is always a scaled-up version of what it was years ago. This
property, however, clearly depends on various kinds of simplifying assumptions, posed primarily for analytical tractability of the model. Among such
simplifications, the most crucial is the fact that many models have ignored
relationships between goods, or at least treated them in a separable way.
In a situation where new goods are continually being introduced, sucha treatment means we ignore two important aspects of economic growth.
First, how consumers value a certain new good depends on the whole bundleof goods they are currently consuming. Thus, the current availability ofgoods aﬀects the prices of new goods when they are introduced.1 Second,
production of a new good depends to a greater or lesser extent on various
kinds of knowledge in the public domain. A large portion of such knowledge
was in fact obtained through the past experiences of private firms and has
subsequently become publicly available. Therefore, the set of goods that has
been produced in the whole economy determines the current distribution of
specific knowledge, and thus aﬀects the production cost of the next goods to
be introduced.
In these two ways, interactions between new and existing goods aﬀect
the kinds of new good introduced by profit maximizing private firms andthe time of their introduction. Moreover, there also exist similar interactions
among existing goods, which will cause a dynamic evolution of their prices
and quantities, and hence aﬀect economic growth rate.
1

For ease of description, we suppose in this paper that consumers directly purchase
diﬀerentiated goods, although it is more common to assume there is a final good producer
who demands diﬀerentiated intermediate goods. Essentially, it is a matter of convenience
to take either view, and it is possible to reinterpret our model in terms of intermediate
goods.

−1−

In this paper, we explicitly introduce these interactions into an endogenous growth model and investigate how the conclusions must be modified.
More specifically, in our setting, which will be formally introduced in the next
section, there are many firms competing monopolistically among themselves,and each of them is allowed to choose the characteristic of its own diﬀerentiated good from an unbounded product space, represented by the realaxis. Here, two opposite forces come into play. On the one hand, since eachconsumer desires to satisfy various needs by purchasing a diverse range ofgoods, every firm has an incentive to choose characteristics that are matchedpoorly by other existing goods. This works as a centrifugal force on the
product space. On the other hand, we assume that knowledge accumulated
by learning-by-doing spills over ‘nearby’ characteristics, and thus production
costs are lower for those goods that have characteristics closer to existing
ones. This works as a centripetal force and these two forces together represent the interaction between goods described earlier.
Analyses in the following sections will show that firms tend to agglomerate
at several endogenously determined points in the product space.2 We would
like to interpret each mass point as an ‘industry’, since within each point every firm produces a mutually symmetric, though diﬀerentiated, good.3 Alongthe path of economic growth, the number of diﬀerentiated goods, or firms, ineach industry varies continuously. This growth process is simply a multisector
extension of the usual endogenous growth models. There is yet another typeof dynamic process: the structure of the economy itself changes discretely by
the emergence or disappearance of industries. Moreover, the timing of emergence and the characteristics of new industries are endogenously determined,
dependent on consumer preferences and production technology.
2

In this paper, we use the term ‘agglomerate’ to mean ‘gravitate towards a common
point in the product space’. Note that it do not mean ‘merger’ of firms in the takeover
sense.
3

Alternatively, we may well call each point a ‘product group’. Indeed, these two ideas
are equivalent in our model since each firm produces only one good. However, in the real
economy where joint production is possible, such an equivalence does not necessarily hold.
We are not concerned here with this point, however.

−2−

In this situation, how will such events aﬀect economic growth? We will
confirm an intuitive property that recently-emerged new industries explain adisproportionately large portion of economic growth. This property implies
that the growth rate will be aﬀected dramatically when industrial structurechanges. In fact, we will show that after a new industry emerges, the overall
economic growth rate sharply declines, at least in the short term, since a
slowdown of knowledge accumulation in currently large industries has a large
negative eﬀect on the growth rate. This might be a surprising result since wewill also see that in the long run, positive economic growth is not sustainable
unless new industries are introduced. As a result, the growth rate persistently
fluctuates in parallel with the sporadic emergence of new industries.4
There is a large body of literature on product diﬀerentiation. As shownin the review by Perloﬀ and Salop (1985), two basic formalizations of product diﬀerentiation have been explored in detail: the Hotelling (1929) spatialcompetition model, and the representative consumer model often associatedwith Chamberlin (1933). Most current endogenous growth models are basedon the representative consumer model, in particular on the work of Dixit andStiglitz (1977), where a representative consumer (or a final good producer)
purchases many goods, varying the proportions of each according to theirprices and exogenously given utility weights. Although this formalization
has a desirable property of permitting multibrand competition, it does notclearly focus on the characteristic of each good nor the relationship between
them. In contrast, Kim (1989), and Kim and Mohtadi (1992) are among the
few studies in the endogenous growth literature that are based on the spatial
4

Hornstein and Krusell (1996) and Greenwood and Yorukoglu (1997) reported the ‘productivity slowdown puzzles’, which attribute productivity slowdown to the learning costassociated with new technologies. This paper oﬀers an alternative explanation to their
hypothesis, and can be viewed as an integration of the puzzles into a theory of cyclical
growth. In fact, similar studies already exist. Helpman and Trajtenberg (1998) presented
a model in which exogenous sporadic improvements in the GPT (General Purpose Technology) caused recurrent cycles in the growth rate. Matsuyama (1999) is a recent workthat focused on the tradeoﬀ between expansion of variety and the growth rate, in thediscrete time setting. In our model, the pattern of introduction of new technologies is
endogenously determined in the continuous time setting.

−3−

competition model. Indeed, they explicitly focus on the good’s characteristic, but competition reduces to a localized phenomenon for them, since eachconsumer purchases only one or two most preferred goods.
We will integrate these two approaches by introducing a two-dimensional
product space. Initially, we will concentrate on a monopolistically competitive economy where each firm can ignore its impact on, and hence reaction
from, other firms. In such a situation, Hart (1985) shows that there must be
at least one dimension of product diﬀerentiation where the ‘non-neighboring
goods property’ holds; that is, goods must be suﬃciently distinctive that eachgood is equally far apart from every other good and thus there is no naturalordering over them. In fact, the Dixit-Stiglitz style competition satisfies thisdesirable property, which can be interpreted in our context as one dimensionof competition that represents the diﬀerentiation ‘within’ each industry. We
will combine this dimension with another of the Hotelling-style spatial competition that corresponds to the diﬀerentiation ‘between’ industries. Thishybrid specification enables us to focus on each good’s characteristic while
retaining features of multibrand monopolistic competition.5
As an engine of economic growth, we will exclusively concentrate on‘learning-by-doing’, the incidental process of knowledge accumulation that
accompanies the production of goods.6 Among various aspects of the learningby-doing process, our model incorporates two important characteristics pointed
out by Young (1991, 1993). First, there appear to be substantial spillover
5

Technically, our formalization of consumer preference is closely related to recent models in urban economics, such as in Fujita, Krugman and Venables (1999), where spatial
structures of cities are analyzed in the context of monopolistic competition. However,there still remains a crucial diﬀerence in that our model has a representative consumer
who actively determines the distribution of her expenditure over the product space, whilein urban economics consumers themselves are distributed locationally.
6

Of course, investment in knowledge often precedes production, for example in thecase of R&D, and many workers invest actively in their human capital. Although theseprocesses are also important, there is empirical evidence that supports our specification
as a close approximation. Jovanovic (1995) reports that even the most advanced countries
spend far more on adoption of existing technologies than on inventing new ones, with his
rough estimate in the U.S. that adoption costs outweigh invention costs by 20 or 30 to 1.

−4−

eﬀects in the development of knowledge across industries, since many of thetechnical and managerial advances brought about by experience in the production of certain goods seem to have applications elsewhere. Thus, productivity increases in each industry are not only a consequence of productive
activity in that industry, but also the result of spillovers from learning-bydoing in other industries. Our model focuses also on the varying degrees ofdiﬃculty of application by formulating the magnitude of the spillover eﬀect
as a decreasing function of the characteristic distance between industries.7
The second empirical regularity relevant to us is the existence of strong
diminishing returns in the learning-by-doing process.8 In particular, we suppose that learning-by-doing is ‘bounded’, which means that even if production experience accumulates unboundedly, the production cost of a particular
good does not fall below a certain amount. Although the existence of sucha bound is an unresolved empirical matter, it has an intuitive implication:
positive economic growth will not persist unless new goods are continually
introduced. Thus, in our model, the learning-by-doing process within anindustry aﬀects the growth rate only transitorily, while its spillover eﬀect
contributes to economic growth in the long run by paving the way for the
emergence of ever newer industries.9
The rest of this paper is organized as follows. After presenting the modelin the next section, we go on in Section 3 to investigate the market equilibrium at each instant for fixed knowledge and population. Dynamic evolutionof the system of industries is analyzed in Section 4, utilizing results fromnumerical simulations. Then, the fifth section clarifies why the growth rate
7

Stoky (1988) presents an alternative formulation of the spillover of knowledge in amore general setting.
8

For extensive references, see Young (1991, p371-72).

9

In this respect, our model contrasts clearly with and is complementary to the recent ‘hybrid’ endogenous growth models presented by Young (1998), Peretto (1998), andDinopoulos and Thompson (1998), which combine the variety expansion model and thequality ladder model. In their models, the primary determinant of long-term growth isquality improvement within each industry, while the expansion of variety is treated as atransitory adjustment process that does not aﬀect the long-term growth rate.

−5−

has cycles. Finally, Section 6 briefly oﬀers an extension that does not require
population growth.

2

The Model

In this section, we develop a specific model that explicitly focuses on the relationships between goods. The first subsection formally defines the productspace and consumer preference on it, under which firms have an incentive to
move apart from each other. The second subsection then specifies production technologies and the processes of learning-by-doing, which induce firms
to agglomerate with each other.

2.1

Preference and Product Space

Assume that the economy has many identical consumers, and that the measure of them, Nt , grows at an exogenous positive rate, λ. Here, time iscontinuous, but in the following, we will omit the time subscript unless it isnecessary. Each consumer has potentially unbounded ‘needs’, represented by
the real axis R ≡ (−∞, ∞), from which she can attain utility. In particular,
let us define a subutility v(r) for each r ∈ R, which represents the level ofsatisfaction of the need, r. Then, overall instantaneous utility is
U=

∙Z ∞

v(r)

β−1
β

−∞

dr

β
¸ β−1

,

(1)

where β is the elasticity of substitution between needs. We assume β > 1,
which means that consumers desire to satisfy an increasing range of needs.
There is a continuum of firms and we index each of them by i ∈ [0, n̄].
Here, n̄ is the measure of the total number of firms, whose evolution will
be determined in equilibrium. Each firm produces one diﬀerentiated good,which can be arbitrarily divided to satisfy various needs. We assume that
consumers benefit from consuming a large variety of diﬀerentiated goods tosatisfy each need. Goods are not necessarily symmetric with each other,
however; that is, each good has its own ‘characteristic’ that represents what−6−

need it matches best. We formally state these considerations by introducing
a two-dimensional product space.
The first dimension, which we call the ‘variety dimension’, enables goods
to be intrinsically diﬀerentiated from each other in a way similar to that
introduced by Dixit and Stiglitz (1977). In particular, we suppose that, fora given fixed budget, the subutility of a need becomes higher when a greaternumber of distinct goods are consumed to satisfy it. In this case, a slight risein the price of a certain good will not eliminate its demand since consumers do
not want to give up this good altogether, as long as no pair of firms produce
the same good. In fact, we will be able to confirm in the next subsectionthat every firm produces a distinct good, and therefore the firms competemonopolistically among themselves.
The other dimension has a spatial structure represented by the real axis
R. We would like to call the latter the ‘need dimension’, since there is a
one-to-one relationship between this dimension and the space of needs, also
represented by the real axis: when a consumer uses a marginal unit of agood that has a characteristic s ∈ R in the need dimension to fulfill a needr ∈ R, the contribution from this marginal consumption to the subutility of
the need r is negatively related to the distance between r and s. This specification parallels Hotelling (1929), and combined with the variety dimension,
opens up the possibility of an ‘industry’ where many firms compete among
themselves for the same need.
Let l(i) ∈ R denote firm i’s choice of its characteristic in the need dimension10 and let c̃(r, i) denote the density of consumption of firm i’s productthat is used by the representative consumer to satisfy need r. Then the
subutility of need r is
σ
∙Z n̄
¸ σ−1
¡
¢ σ−1
−τ |r−l(i)|σv(r) =c̃(r, i)e
di
,
(2)
0

where σ is the elasticity of substitution between goods, and τ the coeﬃcientthat measures how rapidly a good becomes unfit for the need r as its char10

In the following, we will say ‘the characteristic of a firm’s product’ and ‘a firm’s
characteristic’ interchangeably, since each firm produces only one kind of good.

−7−

acteristic l(i) becomes more distant from r. Substituting (2) for (1), we now
have an instantaneous utility function in terms of the list of characteristics
of firms’ products l(·) and the consumption density c̃(·, ·).
Note that, from the consumer’s viewpoint, taking l(·) as given, this is a
two-stage CES utility function of c̃(·, ·). Here we assume that the elasticity
of substitution between goods within a certain need, σ, is higher than thatacross needs, β. This is equivalent to supposing that the marginal utility of
consuming a good to satisfy a certain need r decreases as the subutility ofthat need increases. Under this assumption, each firm has an incentive tochoose a characteristic that is matched poorly by the other existing goods.
It is this centrifugal force that is one of the two major driving forces in thiseconomy.
By making use of the assumption σ > β > 1, we can characterize theutility function in a more convenient manner. Since the instantaneous utility
function is strictly quasi-concave,11 if two firms have chosen the same characteristic, the demands for the products of these two firms would be the sameprovided that both firms have set the same price. In fact, this will be thecase in equilibrium, since both firms confront the same demand curve andthe same cost condition. Thus, instead of the index of each firm, i, we can
express the consumption density in terms of the characteristic of that firm, s.
Formally, let c(r, s) denote the value of c̃(r, i) if there exists any firm i whosecharacteristic l(i) coincides with s. If there is no such firm, let c(r, s) = 0.
Then, (2) can be rewritten as
σ
∙Z ∞
¸ σ−1
¡
¢ σ−1
−τ |r−s|σv(r) =c(r, s)e
dn(s)
,
(20 )
−∞

where n(·) is a distribution function of firms, which is defined as the measureof firms whose characteristics are equal to or to the left of s.
Before closing this subsection, let us consider labor supply and saving
behaviors. First, we assume that each consumer inelastically supplies one
unit of labor and the wage rate is normalized to unity. We also suppose that11

By the assumption σ > β > 1, we have both (σ−1)/σ < 1 and (σ/(σ−1))((β−1)/β) <1, which together establish the strict quasi-concavity of U .

−8−every good is perishable and cannot be stored. In addition, as will be seenin the following, there is no opportunity to invest. Thus, in this economy
where consumers are homogeneous, the credit market involves no trade, andevery consumer spends all of her income at each instant. This implies that
she maximizes the instantaneous utility (1) with (20 ) under the instantaneousbudget constraint
Z ∞q(s)p(s) dn(s) = 1,
(3)
R∞

−∞

where p(s) and q(s) ≡ −∞ c(r, s) dr are the price and the amount of purchase,respectively, of each good that has the characteristic s.

2.2

Learning-by-Doing and Production Technology

We now turn to describe the production side. In this economy, there isneither any kind of entry deterrence nor lump sum setup/exit costs, andevery firm can choose its product’s characteristic freely. At each instant,every firm simultaneously determines whether to enter the market or not, its
characteristic, quantity and price to maximize its profit.
There are two factors of production: labor, and knowledge specific toeach characteristic. We assume that each specific knowledge accumulates bylearning-by-doing processes and has an inappropriable and nonrival nature;
therefore, each firm takes it as given.
Since one kind of input factor is publicly available, each firm hires only
the remaining factor, namely, labor, at the prevailing wage rate. In particular, when a firm produces Q of its good with a characteristic s, the labor
requirement for this firm is
L = (M Q + F )a(s).

(4)

In the above equation, M and F are the marginal and fixed flow cost of
production, respectively, in terms of eﬃciency units of labor. The last term,a(s), represents the number of workers required to generate one eﬃciency
unit. We assume that this requirement depends on the good’s characteristic−9−and is negatively related to the knowledge accumulated with respect to it.
Specifically, we pose a simple relationship between the unit requirement andthe accumulated knowledge,a(s) = 1 +

1
,
K(s)

(5)

where K(s) is the accumulated knowledge with respect to a characteristics. Note that, although eﬃciency units per person, 1/a(s), increase as theknowledge accumulates, there is an upper bound of unity. In other words, aslong as the economy operates on the same set of characteristics, the learningby-doing eﬀect is bounded.
The formula (4) also expresses the total cost of production, since the wage
rate is normalized to unity. From the fact that the average cost decreases withthe production amount, we can confirm that every firm produces a distinct
good, at least in the variety dimension. Suppose that there are two firms
producing exactly the same good and each of them runs with zero profit.
Then, if one of these firms decides to produce another variant in the variety
dimension while keeping its price and characteristic in the need dimension
unaﬀected, this firm will attract twice the demand. In that case, this firmmust attain positive profit, since its average cost has fallen because of theincrease in demand. Thus, such a situation contradicts the assumption offree entry. Also, in a similar way, we can show that no pair of goods are thesame in the variety dimension even though they have distinct characteristics
in the need dimension.
Finally, we must specify the process of accumulating knowledge by learningby-doing. We assume that knowledge of a particular characteristic is proportional to the discounted sum of the past production experience of goods
that have characteristics ‘near’ to the knowledge concerned. Specifically, weformulate the accumulation process as
Z ∞0K̇(s) = γe−ν|s−s | Q(s0 ) dn(s0 ) − δK(s),(6)−∞

where Q(s) denotes the amount that each firm with characteristic s produces.
There are three positive parameters in the above formula: γ captures the− 10 −

overall eﬀectiveness of learning-by-doing; δ represents the depreciation rateof knowledge; and ν is the coeﬃcient of the spillover eﬀect of learning-bydoing across characteristics. Note that ν represents how rapidly the spillover
eﬀect diminishes as the characteristic distance increases.12 We assume in thefollowing that the spillover is not too strong: specifically, ν > ((σ − 1)/σ)τ .
This assumption is fully satisfied when applying some knowledge to anothercharacteristic is no easier than using some good for another need.
This diminishing spillover eﬀect gives firms an incentive to agglomeratewith each other over time. When there is a characteristic chosen historicallyby many firms, the accumulated knowledge with respect to this characteristic
must be greater than neighboring ones. Since the latter implies low production costs for this characteristic, current firms have an incentive to choose it
again. Together with the static centrifugal force arising from consumer preference, this temporal centripetal force plays a central role in the economic
dynamics.

3

Instantaneous Equilibrium

Before turning to dynamic analyses, this section clarifies static propertiesof the market equilibrium, taking as given the values of the state variables
K(·) and N. The first subsection characterizes the instantaneous equilibriumwith a set of conditions, or, alternatively, with two curves representing production costs and demand prices, respectively. After examining global andlocal properties in the second subsection, the third subsection establishes the
optimality, uniqueness and existence of the instantaneous equilibrium, which
are needed to justify numerical procedures.

3.1

Equilibrium Conditions

Let us start by deriving the demand function of consumers for each good withvarious characteristics. To solve a constrained maximization problem of (1),12

In particular, ν → ∞ corresponds to the case where there is no spillover.

− 11 −

(20 ), with (3), we use a two-step method. In the first step, we maximize
the subutility of each need under a given expenditure density and derive
the indirect subutility function. Then, in the second step, we maximize theoverall instantaneous utility function with respect to the expenditure density
under the instantaneous budget constraint.
The first problem is to maximize the subutility of each need v(r), definedby (20 ), under the constraint
Z ∞p(s)c(r, s) dn(s) = y(r),(7)−∞

with respect to c(r, ·). Here y(·) is the density function of expenditure, whichat this point we take as given. For each r ∈ R, we can calculate the solution
to this problem and the maximized value, respectively, as13c∗ (r, s; y(r)) = y(r)P (r)σ−1 p(s)−σ e−(σ−1)|r−s| ,v∗ (r; y(r)) = y(r)/P (r),

(8)
(9)

where P (r) is the ‘price index’ of the need r:
P (r) =

∙Z ∞

−∞

1
¸− σ−1¡¢
τ |r−s| −(σ−1)
p(s)e
dn(s)
.

(10)

Next, consider the second problem. Substituting the indirect subutility
function (9) for (1), we can express the instantaneous utility in terms of theexpenditure density, y(·). The problem here is to maximize this utility funcR∞tion subject to the instantaneous budget constraint −∞ y(r) dr = 1. We cancalculate the optimal expenditure density and the maximized instantaneous
utility, respectively, asy ∗ (r) = P̄ β−1 P (r)−(β−1)
U ∗ = 1/P̄ ,
13

(11)
(12)

On the subset of the characteristic space where no firm operates, let us define p(s) =∞ so that the optimal value of c(r, s) becomes zero. In fact, this treatment might beredundant, since on such a subset the value of c(r, s) is relevant to neither the subutility(20 ) nor the budget constraint (3).

− 12 −

where P̄ is the ‘average price index’ over all needs:141∙Z ∞¸− β−1−(β−1)
P̄ =
P (r)
dr
.

(13)

−∞

Collecting these two steps, we can derive each consumer’s actual demand.
Substituting (11) into (8), the optimal consumption rule turns out to bec(r, s) = P̄ β−1 P (r)σ−β p(s)−σ e−(σ−1)τ |r−s| .

(14)

The amount of each good purchased can also be calculated by integrating its
use over all needs:
Z ∞
β−1
−σ
q(s) = P̄p(s)P (r)σ−β e−(σ−1)τ |r−s| dr.
(15)−∞

With this result in hand, we now turn to investigate firms’ profit maximization behavior. Since there is a continuum of firms, any single firm’s
behavior will not change price indexes P (r) or P̄ and thus they take theseindexes as given. Here, we again use a two-step method to maximize a firm’s
profit. Specifically, in the first step, we choose a price-quantity pair subject
to the consumer’s demand function to maximize the firm’s profit, taking its
product’s characteristic as given. Then, in the second step, comparing theresult of the previous step, we find out which characteristic yields the greatest
profit.
Consider a firm that has determined its product’s characteristic to be s.
Even though there already exist some other firms with the same characteristic, this firm has the power to determine its own product’s price, since every
firm selects a distinct good in the variety dimension of the product space.
However, there is a tradeoﬀ between sales quantity, Q, and its price, p. Multiplying the individual demand function (15) by the number of consumers,
this tradeoﬀ is expressed as15Z ∞β−1 −σQ = N P̄ p
P (r)σ−β e−(σ−1)τ |r−s| dr.
(16)
−∞

14

Alternatively, we can interpret P̄ as the ‘expenditure function’ required to attain aunit of instantaneous utility.
15

Note that the equation (15) has been derived from the reduced form of the subutility
function (20 ), where every firm with the same characteristic chooses the same price. Here,− 13 −

Taking its characteristic and the prices of other firms as given, this is a
simple function of its own price, with a constant elasticity σ. Thus, under
the production technology (4), which exhibits constant marginal cost, it isstraightforward to calculate the optimum pricing rule and the maximized
profit:σ
M a(s),
(17)
σ−1
Z ∞
(σ − 1)σ−1∗
β−1
π (s) = σ
N P̄P (r)σ−β e−(σ−1)τ |r−s| dr − F a(s).
σ (M a(s))σ−1−∞p∗ (s) =The next step is to find out which characteristic yields the greatest profit.
By the assumption of free entry, however, there must be no opportunity inequilibrium to attain positive profit. This implies that the whole characteristic space, R, can be divided into two subsets: in one subset every operating
firm earns zero profit, whereas in the other there is no operating firm sincepotential profits are negative. Put formally, the free entry equilibrium requires π ∗ (s) ≤ 0 for all s ∈ R with equality on the support of n(·). Using
the results of the previous step, the condition π∗ (s) ≤ 0 reduces to∙(σ − 1)σ−1
N P̄ β−1
σ
σ−1
σ M F

Z ∞

σ−β −(σ−1)τ |r−s|

P (r)

−∞

e

dr

¸ σ1

≤ a(s).

(18)

Although the above is a key expression in our model, it may seem somewhat messy. However, if we choose units of measurement appropriately, it
simplifies to some extent. First, since our instantaneous utility function exhibits homogeneity of degree in the consumption density, we can freely choose
measurement units for output quantity. Let us choose units in such a waythat the marginal requirement of eﬃciency units of labor, M , coincides with(σ − 1)/σ. Second, our instantaneous utility is also a homogeneous functionin the distribution of firms. Thus, without losing generality, we can freely
choose the unit for the measure of firms. In particular, we choose the unit
however, we want to let each firm choose its price independently of the other firms withthe same characteristic. Thus, to be precise, the demand function (16) must be derived
from the original definition of the subutility function (2) instead of the reduced form (20 ).

− 14 −

so that the flow fixed requirement satisfies F = 1/σ.16
These normalizations simplify the optimal pricing rule (17) and the zeroprofit condition (18) into p∗ (s) = a(s) and p̂(s; n(·), p(·), N ) ≤ a(s), respectively. In the latter expression,∙p̂(s; n(·), p(·), N) ≡ N P̄

β−1

Z ∞

−∞

P (r)

σ−β −(σ−1)τ |r−s|

e

dr

¸ σ1

(19)

represents the price at which a marginal firm with the characteristic s can sella unit quantity, when the distribution of the existing firms, their prices andthe population are given by n(·), p(·) and N , respectively. In the following,
we refer to this value as the ‘unit demand price’. Note that the unit demand
price is expressed explicitly as a function of n(·) and p(·) in addition to N,
because P̄ and P (·) in the RHS of (19) depends on n(·) and p(·).
Substituting the optimal pricing rule and the zero profit condition into
the demand function (16), we can see that each operating firm’s optimal output quantity is always unity. In light of this result, the free entry condition,p̂(s; ·) ≤ a(s) with equality on the support of n(·), has a natural interpretation. In equilibrium, every firm sells optimally a unit quantity at the priceof a(s). This assures zero profit, since a(s) is exactly the cost of producing
a unit quantity of a good with the characteristic s.17 However, if the unit
demand price is below the unit cost a(s), it will be impossible to operate
without deficits. Thus, there is no operating firm with characteristics suchthat p̂(s; ·) < a(s) holds. Conversely, suppose that there is some characteristic where the unit demand price exceeds the unit cost a(s). Then, there
must be an opportunity to acquire positive profit, contradicting the free entry
condition. Thus, p̂(s; ·) ≤ a(s) must hold for all s.
16

If we double the whole distribution of firms and simultaneously halve the output
of each firm and the fixed requirement, the instantaneous utility will be multiplied by21/(σ−1) , whereas all the remaining equations hold without any substantial change. For
this kind of normalization, see Fujita, Krugman and Venables (1999, Chapter 4).
17

Remember that the total labor requirement for producing a unit quantity of a goodwith a characteristic s is (M + F ) a(s), which is equal to a(s) because of the normalizationsof M and F .

− 15 −

Now we are in a position to summarize the instantaneous equilibriumof this economy, in terms of the distribution of firms n(·), their prices p(·)
and the consumption density c(·, ·). Note that the distribution of knowledge,
K(·), has been determined by the past history of the economy, and must
be taken as given at each instant. Formally speaking, the instantaneous
equilibrium {n(·), p(·), c(·, ·)} is a function of K(·) and N .18 The conditions
to be satisfied in equilibrium are the following:(E1) Free entry: p̂(s; n(·), p(·), N ) ≤ a(s) for all s ∈ R with equality on the
support of n(·).
(E2) Monopoly pricing: p(s) = a(s) on the support of n(·), and p(s)= ∞otherwise.
(E3) Consumption rule: for all (r, s) ∈ R2 , c(r, s) is determined by (14).
(E4) The labor market clears:¸Z ∞∙ Z ∞Nc(r, s) dr + F a(s) dn(s) = N.
M
−∞

(20)

−∞

There are two points to note. First, when the conditions (E1)-(E3) aresatisfied, the markets for every diﬀerentiated good clear. In this case, thelabor market also automatically clears by virtue of Walras’ law. Thus, infact, only the three conditions (E1)-(E3) must be confirmed.
Second, the conditions (E1)-(E2) might at first blush suggest that n(·)
and p(·) must be simultaneously determined, since the unit demand priceappearing in (E1) depends both on n(·) and p(·). This is, however, not thecase. In fact, the unit demand price does not depend on the whole profile
of prices p(·), but only on the prices of operating firms, that is, the valuesof p(·) on the support of n(·), which are equal to the known values of a(·).
Thus, we are able to replace (E1) with a simpler version:(E10 ) Free entry: p̂(s; n(·), a(·), N) ≤ a(s) for all s ∈ R, with equality on the
support of n(·).

− 16 −

Figure 1: An example of the instantaneous equilibrium
Condition (E10 ) says that the equilibrium distribution of firms, n(·), is
determined in such a way that the resulting curve of p̂(s; n(·), a(·), N) does
not go above the curve of a(s) and necessarily touches it on the support ofthe distribution. Figure 1 gives an example of these two curves that together
constitute an instantaneous equilibrium. Once we find the equilibrium distribution of firms, the remaining part of the equilibrium, p(·) and c(·, ·), canbe calculated straightforwardly, using the conditions (E2)-(E3).

3.2

Global and Local Characterizations

Unfortunately, the key condition (E10 ) cannot be solved explicitly, except forsome special cases. Nonetheless, analytical properties of the two curves, theunit demand price curve and the unit cost curve, enable us to characterize theinstantaneous equilibrium both from global and local aspects. Specifically,
we investigate in this subsection two important properties of the equilibrium
distribution of firms: the boundedness of the distribution’s support in a static
sense and the possibility of mass points that tend to persist for some time
period. More fundamental properties, such as uniqueness and existence, will
18

Note that a(·) is also exogenous, since it has a one-to-one relationship to K(·).

− 17 −

be treated in the following subsection.
First, let us focus on the global side. We have defined the characteristic
space of goods to be an unbounded real axis, and thus, in principle, firms
can choose characteristics infinitely distant from others to avoid competition.
This consideration suggests that the equilibrium distribution of firms might‘explode’ in the characteristic space. In such an event, however, the unit
production cost of goods would also rise unboundedly, since the economy has
accumulated relatively little knowledge at distant characteristics. In fact, we
can show that such an explosion never occurs, as long as the economy has
never experienced such a phenomenon before in its whole history. Formally,
we will prove the following: if the support of nt0 (·) is contained in some finite¯ ⊂ R for all t0 < t, then, another finite interval I 0 ≡ [I 0 , I¯0 ]interval I ≡ [I, I]
exists such that the support of nt (·) is contained in I 0 .
To prove this, let us examine the slope of the unit demand price function.
It turns out that this function is smooth and we can calculate its derivativewith respect to s:
Z ∞∂ p̂(s; ·)
σ−1
1−σ
=
τ p̂(s; ·)sgn(s − r)P (r)σ−β e−(σ−1)τ |r−s| dr,∂s
σ−∞where we have omitted arguments other than s in p̂(s; n(·), a(·), N). By
straightforward comparison of (19) and the above, we can show that there isan upper bound on the slope of the unit demand price function that depends
only on the parameters:¯¯¯ 1 ∂ p̂(s; ·) ¯ σ − 1¯¯(21)¯ p̂(s; ·) ∂s ¯ < σ τ.

This result can easily be interpreted: to satisfy a certain need, consumers
can substitute for some good another ‘poorly matched’, but inexpensive one,with an extra ‘unfitness’ cost that increases exponentially with the distancein the characteristic space; thus, the unit demand price curve cannot haveslopes larger than the value that corresponds to this cost.
We utilize this fact to prove the claim. Suppose that the right-hand end ofthe ‘historical frontier’ has not gone beyond the characteristic I¯ by time t.1919¯ ∞) for all t0 < t.
That is, there is no support of nt0 (·) in (I,− 18 −

Then, we see from (6) that for characteristics currently ‘beyond the frontier’,¯ ∞), the accumulated knowledge monotonically decreases with s ats ∈ (I,
the rate of ν. This can also be expressed in terms of production cost:¡¢¯¯ − 1 eν(s−I)¯
at (s) = 1 + at (I)for all s ≥ I.
(22)
On the other hand, the condition (E10 ) guarantees that a local inequality¯ ≤ at (I)¯ always holds in equilibrium. Combining this inequality with thep̂(I)upper bound for the slope (21), we have an upper bound for the unit demand
price curve beyond the historical frontier:¯
τ (s−I)¯ σ−1σp̂(s; ·) ≤ at (I)e¯
for all s ≥ I.

(23)

Let us compare these two expressions. When the accumulated knowledge
at the frontier characteristic is no more than ν/(ν − ((σ − 1)/σ)τ ), we cansee that the upper bound curve for the unit demand price, given by the¯ In thisRHS of (23), stays below the unit cost curve (22) for all s ≥ I.
case, the historical frontier does not expand since it cannot be profitableto produce goods with characteristics beyound the current frontier. Even if
this is not the case, the unit cost must eventually exceed the unit demand
price for a suﬃciently large s, from the assumption ν > ((σ − 1)/σ)τ . In
¯ we can always find a finite
fact, given values for the parameters and a(I),value I¯0 such that the unit cost curve (22) strictly dominates the RHS of(23) for all s > I¯0 , which in turn implies that there must be no support ofnt (·) beyond I¯0 . Obviously, the same argument can be made for I and I 0 .
Thus, we have established that the instantaneous equilibrium distribution isbounded, provided that the whole history has been bounded.20
Next, we turn to the local property. Note that our model deals directlywith the distribution function of firms n(·), rather than the more analyticallyconvenient density functions. In fact, this somewhat awkward treatment
is indispensable, since mass points are likely to emerge in the equilibrium
distribution of firms. Moreover, once such points have emerged, they usuallypersist for some time period.
20

Note that this is a static property. In a dynamic sense, as we will see in the next
section, the historical frontier expands without bounds.

− 19 −

To illustrate these points, it is useful to clarify the curvature of the unit
demand price function. For ease of description, let us diﬀerentiate p̂(s; ·)σtwice, rather than the function itself. This yields∂ 2 p̂(s; ·)σ= −2(σ − 1)τ N P̄ β−1 P (s)σ−β + (σ − 1)2 τ 2 p̂(s; ·)σ .
(24)∂s2
The first term in the RHS shows that the curvature at a certain characteristic
is larger when the price index at this characteristic, P (s), is lower. Under
the equilibrium condition (E2) and the definition of the price index (10), theterm simply states that the curvature becomes larger when there is a greaternumber of firms operating near this characteristic. Even though a mass offirms may have agglomerated, however, there is also an upper bound on
this curvature: because the first term cannot become positive, the curvaturecannot exceed the value of the second term in the RHS, which is no greaterthan (σ − 1)2 τ 2 a(s)σ under the equilibrium condition (E10 ).
With this fact in hand, we can now explain why a locally isolated masspoint may emerge in the equilibrium distribution of firms. Assume that
there is an interval I in the characteristics space, on which firms are distributed continuously. Then, the free entry condition assures that the equality a(s) = p̂(s; ·) holds for all s ∈ I. As knowledge accumulates, however,the shape of the unit cost curve continually changes, and it would not be unusual for the curvature of a(s)σ to eventually exceed (σ − 1)2 τ 2 a(s)σ in somesubinterval I 0 ⊂ I. Suppose that this occurs and that some firms still remainin this subinterval. Then, the unit demand price curve necessarily touches
the unit cost curve at only one point in the interval I 0 , since the former has alarger curvature than the latter. Moreover, this locally isolated point in thecharacteristic space generically contains firms with positive measure, since
this is the only characteristic in the interval I 0 with which firms can operate
without deficits.
How, then, can such a mass point persist thereafter? Assume that there
already exists a mass point of firms that has persisted for some time period.
By the process of knowledge accumulation (6), these firms must have createdan upward kink in K(·), which also implies a downward kink in a(·). Since
the unit demand curve is everywhere smooth, it tends to touch the curve− 20 −

of a(s) only on the point of kink, at least locally, rather than continuously
over an interval that contains the kink. Thus, such a locally isolated masspoint tends to persist. Moreover, further accumulation of knowledge by thesefirms further strengthens the kink in K(·), and, again, makes the mass point
sustainable in the future.
In Section 4, we will confirm these global and local properties in a dynamic
context by numerical procedures. Before turning to such analyses, however,
we must establish some more fundamental properties of the instantaneous
equilibrium in the following subsection.

3.3

Optimality, Uniqueness and Existence

Although previous subsections have characterized the instantaneous equilibrium in several ways, it remains to be shown that there actually exists only
one equilibrium for each pair of knowledge, K(·), and population, N. This isparticularly important in the numerical analyses to follow: even if numerical
procedures find a solution, it cannot be assured to occur in reality unless it is
the unique solution. We tackle this issue by focusing on another fundamental
concern, the welfare properties of the instantaneous equilibrium.
Since the processes of accumulation of knowledge are perfectly external
for each firm, it is unlikely that the equilibrium path induces an intertemporally optimal allocation. We can show, however, that the instantaneous
equilibrium is ‘instantaneously optimal’: given the current distribution ofknowledge, it attains the maximal instantaneous utility.21 After proving theexact coincidence of the instantaneous equilibrium with the instantaneously
optimal allocation, we will proceed to show that such an allocation uniquely21

As is well known, Dixit and Stiglitz (1977) have shown that, in a static model with only
one dimension of diﬀerentiation, the monopolistically competitive equilibrium is actually
the first best solution, provided that the utility function of the representative consumer
has a CES form. The instantaneous equilibrium in our model is a generalization of theirmodel, in that if all firms are forced to choose the same characteristic, or diﬀerentiation in
the need dimension is prohibited, the resulting equilibrium reduces to virtually the sameas the Dixit-Stiglitz model. Thus, the first best will result in such a special setting. In thetext, we show that this property indeed extends to our generalized economy.

− 21 −

exists. More precisely, this subsection will proceed in the following way. (i)Formulate the problem of maximizing the instantaneous utility subject tothe current state of knowledge, K(·), and limited workforce, N . (ii) Provethat the solution to the maximization problem is unique. (iii) Show that
any instantaneous equilibrium is a solution to this maximization problem,and vice versa. (iv) Show that such an equilibrium-maximizer always exists.
These procedures establish all of the three claims: instantaneous optimality,
uniqueness, and existence of the instantaneous equilibrium.
The problem to be formulated first is to maximize the instantaneous
utility (1) and (20 ) with respect to the distribution of firms n(·) and theconsumption density c(·, ·), subject to the resource constraint (20). This isa constrained maximization problem, but not a standard one as it includes
maximization with respect to a distribution function. Moreover, because of
the possibility of mass points in the distribution, we cannot reformulate thisproblem using a density function of firms. These facts imply that, although
we want to show equivalence of the solution to the equilibrium, its direct
proof would be exceedingly involved. Rather than attempting this diﬃcult
task, let us introduce a supposedly innocuous restriction on firms’ behavior. Specifically, we discretize the characteristic space, R, into arbitrarily
fine grids and restrict the behavior of firms so that they can select theircharacteristics only from points on these grids.22
Let S = {s1 , s2 , · · · } be a countable set of distinct points in the real axis,
R, from which firms can choose their characteristics. Assume that a small
number ² > 0 exists such that |si − sj | ≥ ² holds for all i 6= j, and thus S
22

Besides the ease of analysis, there are three reason to adopt this method. First, as longas the points in the grids are countable, we can take arbitrarily fine grids. When the grids
are made suﬃciently fine, we can legitimately expect the model not to exhibit significant
diﬀerences from the continuum case. Second, because of the temporal centripetal force
described earlier, a great majority of firms tend to agglomerate in several points in thecharacteristic space, rather than to be distributed continuously. Finally, one of the reasons
why we want to establish uniqueness is to validate the following numerical analyses, wherediscretization is inevitable as long as digital computers are used. Alternatively, it is also
possible to interpret our original model itself as a limiting case of the discrete setting
presented in the text.

− 22 −

has no cluster point. Since firms are distributed exclusively on the points in
S, their distribution can be expressed in terms of a sequence of nonnegative
numbers {n1 , n2 , · · · }, where nj denotes the measure of firms with characteristic sj . With these notations, it would be possible to construct a set ofstandard first order conditions for the instantaneously optimal allocation.
However, there is another subtle issue to be clarified: the first order conditions may not be suﬃcient, since the objective function (1) with (20 ) is
not concave, nor is the resource constraint (20) linear. To resolve this issue, let us transform two sets of variables. First, instead of maximizing the
instantaneous utility U directly, we maximize its increasing transformationŨ ≡ U (β−1)/β . Second, in place of c(·, sj ), which indicates the consumption
density of each good with characteristic sj , we introduce hj (·) ≡ nj c(·, sj ),which represents the total consumption density of all goods with characteristic sj .23 Using these variables, the problem is transformed intoZ ∞β−1
maximizeŨ =v(r) β dr,(10 )∞{nj ,hj (·)}j=1

−∞

where
v(r) =

" ∞
X

1
σ

nj

j=1

¡

¢ σ−1
hj (r)e−τ |r−sj | σ

σ
# σ−1

,

subject to the resource constraint
¶
Z ∞
∞ µ
X
MN
hj (r)dr + F nj a(sj ) = N.
j=1(200 )(200 )−∞

23

Note that, for any characteristic s outside the grids, trivially c(·, s) = 0 holds since wehave assumed that there is no firm with such a characteristic. Therefore, only the valuesof c(·, sj ), j = 1, 2, · · · , need to be specified. Strictly speaking, however, the relationshipbetween c(·, sj ) and hj (·) is not always one to one, since, when nj = 0 and hj (r) > 0 forsome j and r, there is no counterpart of c(r, sj ) to such hj (r). Thus, we unexpectedly
have a broader freedom in this transformed problem. However, this extended freedom will
never be exploited, since, when nj = 0, the marginal benefit of increasing hj (r) is zero,
whereas its cost is positive.

− 23 −

Clearly, the constraint (200 ) now becomes a linear one, and in Appendix A
we prove that Ũ is indeed a concave function with respect to the variables
concerned.
To be precise, there are two other sets of inequality constraints; that is,nonnegativity of {nj } and {hj (·)}. As will soon become clear, the latter neverbinds, whereas the former sometimes does. Thus, we must solve an equality
and an inequality constrained problem. After considerable calculations, theKuhn-Tucker condition for the maximizer of this problem turns out to be
the existence of some number ξ, the Lagrange multiplier for the resource
constraint, such that the following conditions (K1)-(K3) are satisfied.24(K1) First order condition for the distribution of firms: for each j = 1, 2, · · · ,two inequalities, nj ≥ 0 and
∙
¸ σ1
Z ∞
σ−β
σ β − 1 −1
−
ξ
N 1−σ
v(r) β e−τ (σ−1)|r−sj | dr ≤ a(sj ),
σ−1 β
−∞

hold. In addition, the latter holds with equality if nj > 0.
(K2) First order condition for the consumption density: for each j = 1, 2, · · ·and each r ∈ R,
µ
¶σ
σ−β
σ β−1
1
hj (r) =nj v(r)− β e−τ (σ−1)|r−sj | .
σ − 1 β ξNa(sj )(K3) Resource constraint: equation (200 ) holds.
Since all constraints are linear and the objective Ũ is concave, these conditions are both necessary and suﬃcient, and the set of optima is convex.
Moreover, Appendix B shows that the optimum is in fact unique. Combining these facts together, it is assured that the set of variables that satisfies
the conditions (K1)-(K3) is unique.
Once the set of conditions for the optimal allocation is found, we are in
the position to show its equivalence to the equilibrium conditions (E1)-(E4).
24

For details about constrained maximization and the Kuhn-Tuchker condition, see, forexample, Mas-Colell, Whinston and Green (1995, Section M.K).

− 24 −

Before proceeding, however, remember that we have modified the economy
slightly so that firms are allowed to choose their characteristics only fromthe points in S. Thus, to be consistent, we must also modify the free entry
condition (E1) into
(E100 ) Free entry: for each j = 1, 2, · · · , two inequalities, nj ≥ 0 and p̂(sj ; {nj }, p(·), N)≤ a(sj ), hold. In addition, the latter holds with equality if nj > 0.
This is simply a weaker version of the original condition, in that, while (E1)
demands that the free entry condition holds over the whole characteristic
space R, this version requires the condition to hold only over the subset
S ⊂ R, since in this economy firms cannot choose any point outside S fromthe beginning.
Let us start the proof. In one direction, we show that any instantaneous
equilibrium is instantaneously optimal. Suppose that we have an instantaneous equilibrium, {{nj }, p(·), c(·, ·)}, which satisfies the conditions (E100 )and (E2)-(E4). Then, our task is to show that there exists an appropriate
Lagrange multiplier ξ, such that the conditions (K1)-(K3) are satisfied.
First, it is clear that the resource constraint (K3) is equivalent to thelabor market clearing condition (E4). Thus, (K3) is trivially satisfied.
Next, using (9) and (12), we can eliminate price indexes from the equilibrium consumption density (E2). Then, by applying the definition of hj (r),this condition becomes³´
β−1 −σσ−βhj (r) = a(sj )U β
nj v(r)− β e−τ (σ−1)|r−sj | ,which indeed coincides with the condition (K2), provided that we choose themultiplier to beξ=σ β − 1 −1 β−1
N U β .
σ−1 β

(25)

This coincidence means that the equilibrium consumer behavior is in fact
optimal given the distribution of firms, which is a natural outcome since
all consumers have the same preference and there is no contemporaneous
externality.
− 25 −

Finally, eliminating price indexes from the free entry condition (E100 ),and substituting the multiplier (25) into it shows that its equivalence to(K1) is straightforward. This fact shows that the unit demand price function represents the marginal benefit of increasing the number of firms witheach characteristic. Collecting these three results, we have proven that the
instantaneous equilibrium is instantaneously optimal.
In the other direction, we show that the instantaneously optimal allocation constitutes an instantaneous equilibrium. Suppose that there is an
optimal allocation {{nj }, c(·, ·)} and a Lagrange multiplier ξ, which together
satisfy the Kuhn-Tucker condition (K1)-(K3). We now prove that there exists
a set of prices, p(·), such that the equilibrium conditions (E100 ) and (E2)-(E4)are satisfied.
Let us choose a set of prices according to the condition (E2). Then, by
substituting the condition (K2) for (200 ), we can solve for v(r) in terms of ξ,
N and p(·). Once again substituting this result into the conditions (K1) and(K2), we have the following two results:∙Z ∞ξ˜−β N 1−β

P (r)σ−β e−τ (σ−1)|r−sj | dr

¸ σ1

≤ a(sj )(26)c(r, sj ) = ξ˜−β N −β P (r)σ−β p(sj )−σ e−(σ−1)τ |r−s| .

(27)

−∞

with equality if nj > 0 andTo minimize notation, here we have introduced a transformed Lagrange multiplier ξ˜ ≡ (σ/(σ − 1))((β − 1)/β)ξ. Note that the above expressions areequivalent to the equilibrium conditions (E100 ) and (E3), respectively, if andonly if ξ˜ = P̄ −(β−1)/β /N holds. We next show that this is actually the caseunder the resource constraint (K3).
The total labor requirement in the economy, represented by the LHS of(200 ), consists of two parts, the sum of the marginal requirements and thesum of the fixed ones. Let us calculate the amount of each part, given the
˜ Utilizing the equation (27), we can(transformed) Lagrange multiplier ξ.

− 26 −

calculate the first part as
¶
Z ∞
∞ µ
X
¡
¢1−β
N
M
nj c(r, sj )dr a(sj ) = M ξ˜−β N P̄
.
j=1

(28)

−∞

By integrating the equation (27) with respect to r and applying the condition(26), we have the familiar result that each firm produces a unit quantity.
Again, combining this property and (28), the second part becomes
F

∞
X
j=1

¡
¢1−β
nj a(sj ) = F ξ˜−β N P̄
.

(29)

¡
¢1−β 25
These two equations show that the total labor requirement is ξ −β N P̄.
From this expression, we can conclude that the transformed Lagrange multiplier must have the value ξ˜ = P̄ −(β−1)/β /N , at which the labor requirement
coincides with the labor supply, N, and thus the resource constraint (K3)
is satisfied. Therefore, the conditions (E100 ) and (E3) hold in the optimal
allocation.
Finally, the condition (E4) holds by virtue of (K3). This concludes theproof for this direction. Collecting the results for both directions, we have
proven that the instantaneous equilibrium exactly coincides with the optimal
allocation. Moreover, this fact implies that the instantaneous equilibrium is
unique, since we have already shown that the optimal allocation is unique.
The remaining task is to prove the existence of such an equilibrium allocation. In the previous subsection, we showed that the support of theequilibrium distribution is contained in a certain bounded interval, as longas the whole history has also been bounded. Then, there are only a finite
number of points in S that are also contained in the interval. Without losing
generality we can denote these points by s1 , · · · , sJ , where J is a finite number. We assume that this is the case, and refer to the property that nj = 0
holds for all j > J as condition (B).
Note that, since we have established the coincidence of the optimal allocation with the instantaneous equilibrium, the latter also maximizes U subject
25

Recall that we have normalized M and F into (σ − 1)/σ and 1/σ, respectively.

− 27 −

to the resource constraint (E4) and, of course, satisfies (E2), (E3) and (B).
Keeping this fact in mind, consider a constrained maximization of U withrespect to {nj }, c(·) and p(·), subject to (E2), (E3), (E4) and (B). Then, if
a solution to this problem exists, it actually constitutes the instantaneous
equilibrium, since the maximization problem is not altered substantially either by the additional variable p(·) or by the additional constraints (E2),(E3), or (B).
Let us prove the existence of a solution to this transformed problem.
From the constraints (E2) and (E3), the instantaneous utility U can be expressed as 1/P̄ . In addition, we can confirm that the output of each firm isP
unity; thus, the constraint (E4) can be simplified to j a(sj )nj = N . These
conditions also say that, p(·) and c(·, ·) will be passively determined when{nj } is specified. Thus, in eﬀect, this is the problem of maximizing 1/P̄ withrespect to {nj }. By virtue of the constraint (B), moreover, we only haveto deal with J nonnegative variables. Therefore, the problem simplifies toa constrained minimization of P̄ with respect to n1 , · · · , nJ , under the conP
straints Jj=1 a(sj )nj = N and nj ≥ 0 for j = 1 · · · J. This is an intuitive
characterization of the instantaneous equilibrium, in which the distributionof firms minimizes the average price index. It is straightforward to confirm
that the objective P̄ is a continuous function of n1 , · · · , nJ , and that theadmissible set for the control variables is compact. Thus, we are assured ofa solution. This establishes the existence of the instantaneous equilibrium.

4

Dynamic Evolution

Having finished the necessary arguments, we are now in a position to explore dynamic evolution of the economy. Recall that our model includes two
dynamic elements: the process of accumulating knowledge, specified by (6),and exogenous population growth at the rate of λ. As proven in the previous
section, given the current set of knowledge Kt (·) and the current population
Nt , a unique distribution of firms exists that satisfies the key condition (E10 ),which we denote by n̂(·; Kt (·), Nt ). Then, utilizing the fact that every firm

− 28 −

produces unit quantity in equilibrium, we have an autonomous system withrespect to K(·) and N:
Z ∞0K̇t (s) = γ
e−ν|s−s | dn̂(s0 ; Kt (·), Nt ) − δKt (s),(30)−∞

Ṅt = λNt .

(31)

While the above equations fully characterize the evolution of the equilibrium distribution of firms, there is another profound problem: how does thefirst industry emerge? When there has never been any firm, the accumulation process (6) implies that there is no knowledge at all, and (5) states thatany production is prohibited. Thus, it is clear that our model is not at all
suited to answer this kind of problem. Although it is not diﬃcult to add someelements to our model to answer this issue, we would like simply to pose aplausible initial condition for the dynamics, since the aim of this paper is todescribe how systems of industries interact to exhibit various dynamics, but
not to deal with the ultimate origin of them.
As a starting point for the analysis, we adopt an economy with only one
small industry, where every firm chooses the same characteristic.26 Specifically, we assume that the distribution of knowledge at the initial date t = 0
isK0 (s) = K̄0 e−ν|s| ,(32)

where K̄0 is a small positive value. Given (32) and an initial population
N0 > 0, we now construct the initial instantaneous equilibrium.
Substituting (32) for (5), we can derive the unit cost curve:a0 (s) = 1 + K̄0−1 eν|s| .

(33)

This curve is symmetric with respect to the characteristic ‘zero’ and has adownward kink there. If every firm chooses this characteristic, the labor
26

Without losing generality, we can refer to this characteristic as s = 0, since thecharacteristic space, R, is ex ante symmetric.

− 29 −

market clearing condition, (E4), requires the number of firms at the characteristic ‘zero’ to be N0 (1 + K̄0−1 ). Formally, when this is the case, the initial
distribution of firms is expressed as(
0for s < 0,¡¢(34)
n0 (s) =−1for s ≥ 0.
N0 1 + K̄0

Then applying (33) and (34) for the equilibrium conditions (E2) and (E3), we
can determine the whole set of variables {n(·), c(·, ·), p(·)}, which completely
characterizes the instantaneous equilibrium.
However, we must still confirm whether this set of variables truly constitutes an instantaneous equilibrium, by checking the remaining free entry
condition (E10 ). From (33) and (34), we can calculate the price indexes P0 (r)and P̄0 . Substituting these indexes for (19), we have an explicit expression
for the unit demand price function,¡¢p̂(s; n0 (·), a0 (·), N ) = 1 + K̄0−1 Ψ(s),(35)
where Ψ(s) is a symmetric, smooth function that depends only on s andparameters:
"R ∞
# σ1
exp
{(1
−
σ)τ
|r
−
s|
+
(σ
−
β)τ
|r|}
dr
R∞
Ψ(s) = −∞.
(36)
exp {(1 − β)τ |r|} dr−∞

Here, we have to compare the unit demand price curve (33) and theunit cost curve (35). For this purpose, it is useful to define another function that expresses the gap between these two curves at each characteristic,Ω(s; n(·), a(·), N) ≡ p̂(s; n(·), a(·), N ) − a(s). We refer to this function as the‘market potential function’, since it represents the profitability of entering
the market by producing a good with each characteristic. Using this function, we can restate the free entry condition (E10 ) in the following way: themarket potential curve should not exceed zero for all s ∈ R, and must equal
zero on the support of n(·).
From (33) and (35), we can derive the market potential curve for the
one-industry economy:³´¡¢Ω(s; n0 (·), a0 (·), N0 ) = Ψ(s) − 1 − K̄0−1 eν|s| − Ψ(s) .
(37)− 30 −

With this expression in hand, we can examine whether this economy satisfies
the free entry condition. First, let us examine the profitability of the existing
industry, s = 0. Since Appendix C shows Ψ(0) = 1, both expressions inparentheses in the RHS of (37) become zero, which implies exactly zero
market potential for the industry.
On the other hand, what is the profitability of operating outside theexisting industry? Appendix C shows 1 < Ψ(s) < eν|s| for all s 6= 0, whichmeans that the two expressions in parentheses in the RHS of (37) are strictly
positive and negative, respectively. Thus, for suﬃciently small K̄0 (that is,large K̄0−1 ), we can see intuitively that the second term will dominate for alls 6= 0, and thus the condition (E10 ) will be satisfied. Conversely, if K̄0 is not
so small, there may exist some characteristic where the first term dominates,
which implies violation of the condition (E10 ). With such a distribution of
the initial knowledge, the true instantaneous equilibrium diﬀers, in fact, fromthe one-industry economy characterized by (34). In Appendix D, we formally
prove that there is a finite positive constant K̄ ∗ such that the condition (E10 )is satisfied if and only if K̄0 ≤ K̄ ∗ .
In the following, we assume that the initial knowledge is suﬃciently smallthat 0 < K̄0 < K̄ ∗ holds. Then, the initial instantaneous equilibrium involves
only one industry and thereafter knowledge accumulates according to theequation (30). In general, this process involves a continuum of variables, thewhole set of knowledge for each specific characteristic. As long as the oneindustry structure persists, however, we can in fact concentrate on only oneof them, K̄t ≡ Kt (0), since an obvious relationship between them continues
to hold:
Kt (s) = K̄t e−ν|s| .

(38)

Moreover, it turns out that K̄t follows rather simple dynamics,K̄˙ t =γNt
− δ K̄.
1 + K̄t−1(39)

From (39) and (31), it is straightforward to draw a phase diagram in(N, K̄) space, given by Figure 2, and together with initial values of K̄0 and− 31 −

Figure 2: The phase diagram of the one-industry economy
N0 we can see how this economy evolves. Note that, however, this reduced
dynamic system applies only while the economy has the one-industry structure; but for how long does it persist? Since the distribution of knowledge(38) at any time has the same form as the initial distribution (32), we can
examine the equilibrium conditions at each instant in exactly the same way
as we examined the initial equilibrium; that is, the one-industry structure,similar to (34), constitutes the instantaneous equilibrium if and only if K̄tdoes not exceed K̄ ∗ . The phase diagram shows that this structure is boundto become unsustainable in finite time, as long as the initial condition satisfies N0 > 0 and 0 < K̄0 < K̄ ∗ . We denote by t∗ the critical time such thatK̄t∗ = K̄ ∗ holds.
Before time t∗ , the potential curve Ω(s; nt (·), at (·), Nt ) stays below thehorizontal axis everywhere except the origin. This means that it is not profitable for any firm to deviate from the characteristic ‘zero’. In other words,
the centripetal force induced by the learning-by-doing eﬀect globally dominates the centrifugal force induced by the desire of consumers to satisfy awider range of needs. As knowledge accumulates, however, the centripetal
force gradually gets weaker. Because of the spillover eﬀects of learning-by-− 32 −

doing, the unit cost falls not only for the existing industry, but also outside
it, and, moreover, the cost falls more rapidly at somewhat distant characteristics, where there still remains a relatively large gap between the current
cost and the lower bound. Thus, the potential curve moves upward continually away from the origin and eventually, at time t∗ , touches the horizontal
axis at some point s∗ ∈ (0, ∞).27 This means that it would then exactly
pay to enter the market by producing a good with characteristic s∗ , provided
that every existing firm continues to choose the characteristic ‘zero’.28 Then,slightly after the time t∗ , if no firm changes its characteristic from zero, theopportunity of operating around s∗ provides strictly positive profit, contradicting the assumption of free entry. Thus, some firms must change theircharacteristic, which leads to the emergence of new industries.
Numerical Simulation
After the critical time t∗ , the reduced dynamic system (39) and (31) no longer
applies and we must deal directly with the general system (30) and (31).
Since the equilibrium distribution of firms at each instant, n̂(·; Kt (·), Nt ),could not be derived explicitly, it seems sensible at this point to turn tonumerical simulation. To implement such a simulation, two kinds of approximation are necessary. First, since arbitrary distributions of firms on
continuous space, R, cannot be handled numerically, we discretize this spacewith a small fixed interval of 0.01. Second, although the evolution of thiseconomy is characterized by a system of diﬀerential equations, its solution
requires a finite diﬀerence method that in eﬀect replaces a continuous time
problem with a discrete time problem. Specifically, we divide a unit time,
See Appendix D for the proof that such s∗ ∈ (0, ∞) exists. Note also that, since thepotential curve is symmetric around zero, there is another point where the curve touches
the horizontal axis, namely, −s∗ .
27

Strictly speaking, at the very critical time t∗ , no firm will choose the characteristics∗ . At time t∗ , the potential curve, derived from the one-industry distribution, does notexceed zero for all s ∈ R, and thus the one-industry structure nonetheless remains as anequilibrium. Moreover, there is no other possibility, since the instantaneous equilibrium is
unique.
28− 33 −

which can be considered as a year, into 20 subunits and solve a system ofdiﬀerence equations.29
As a benchmark, we select parameter values for the preference of therepresentative consumer to be β = 2, σ = 4 and τ = 1. For the processof knowledge accumulation, we specify γ = 2.4, δ = 0.24 and ν = 1. The
remaining parameter, the population growth rate, is set to λ = 0.024. Finally,
we must specify initial conditions for this economy, K̄0 and N0 . The initialknowledge for producing goods in the initial industry is set to K̄0 = 0.1
so that the initial price becomes a0 (0) = 11. Recall that the lower boundfor the unit cost is unity, and thus there remains a considerable margin forcost reductions because of learning-by-doing within the initial industry. We
select the initial population to be N0 = δ/γ = 0.1, since, if the populationis smaller than δ/γ, the phase diagram, Figure 2, shows that the stock ofknowledge is always decreasing and thus there is no possibility of emergenceof new industries.
Figure 3 depicts the simulated evolution of the distribution of firms. As
the economy always evolves symmetrically around the characteristic ‘zero’,
we show results only for the characteristic s ≥ 0. In particular, black lociin the figure represent the evolution of the support of nt (·), from which we
can read several dynamic properties of the system of ‘industries’.30 First, ateach time after t∗ , the economy consists of a finite number of distinguishable
industries, rather than one big industry that contains a continuous range ofcharacteristics. This in turn implies that structural changes, namely, emergence or disappearance of industries, occur sporadically at distinct points in
29

We have confirmed that these kinds of discretization are robust, in that changing
the units of discretization does not alter the result substantially. Thus, we claim thatnone of the substantial results in our simulation depend upon the discrete nature of theprocedures.
30

Here, it is necessary to define formally what we mean by the term ‘industry’ in ourmodel. At each time t, if the support of the distribution of firms, supp nt (·) ⊂ R, is a
joint set, we say that all firms belong to one industry. If not, we say that each industry iscomposed of a set of firms in a joint subset of supp nt (·). In a dynamic sense, the historyof an industry is identified by a joint subset of the graph of supp nt (·) in (t, s) space.

− 34 −

Figure 3: Evolution of industries (s ≥ 0)time. Second, except for a short time interval after emergence, all firms ineach industry choose the same characteristic, that is, agglomerate in a masspoint. Moreover, once they have agglomerated, there seems to be a strong
hysteresis in their characteristic.31 Third, although some industries disappear in finite time, everlasting industries are roughly evenly spaced and their
number increases without bound in a dynamic sense.32 We also anticipate
from the figure that there is some cyclicality in the evolution of the industrial
structure, although the durations of the cycles seem to become shorter as the
economy becomes larger.
31

During a short time interval after emergence, a newly emerged industry seems toshift its set of characteristics continually in the direction of even newer characteristics.
However, as we have seen in Subsection 3.2, once a mass of firms stays in a point on thecharacteristic space, such a mass point creates a downward kink in the unit cost curve andthereafter tends to persist.
32

This contrasts to the argument in Subsection 3.2, where we proved that the economy
is bounded in a static sense.

− 35 −

The figure also shows the evolution of the size of each industry. Specifically, the height of the gray area above the support of each industry represents
the total revenue, or equivalently, the number of workers, in that industry.
Until the critical time t∗ , the total revenue in the initial industry steadily
increases in parallel with the population growth. After the emergence of acouple of new industries, however, the size of the initial industry begins toshrink and eventually the initial industry is dominated by new ones. There
are two reasons for this. First, since there remains little margin for cost reduction in the initial industry, the prices of goods in new industries fall more
rapidly than those in the initial industry by the learning-by-doing process.
Thus, new industries gradually get a larger share of consumer expenditure.
Second, since two industries have emerged symmetrically, the initial industry
is no longer located on the frontier. While each new frontier industry attracts
a large demand from the needs outside the frontier where other industries
have small competitive powers, the initial industry must compete with otherindustries on both sides. Thus, even if all industries post the same prices,
the frontier industries attract relatively larger demands. When industries
currently on the frontier become suﬃciently large, two more new industries
emerge and a similar process restarts.
Finally, let us explain why some industries disappear. When a frontier industry emerges, it attracts a large demand from the needs beyond its
characteristic. Thus, it may be profitable to produce a new good with acharacteristic not so distant from existing industries. When the next frontier industries emerge, however, the demand that the industry previously on
the frontier can attract significantly falls since it now has a characteristic too
close to other industries on both sides. In some cases, the demand eventually
becomes so small that no firm in this industry can reach zero profit. Such aphenomenon occurs partly because no firm is forward-looking, as knowledge
is inappropriable. Is, then, such an ephemeral industry completely futile fromthe viewpoint of economic growth? In fact, it plays an important historicalrole: the knowledge accumulated by such an industry spills beyond itself andserves as a stepping-stone to new, possibly everlasting, frontier industries.

− 36 −

5

Growth Cycles

In the previous section, the simulation results showed that industries sporadically emerge and disappear as the economy grows. Here, we would liketo investigate how such continual changes in industrial structure aﬀect thepattern of the macroeconomic growth rate. Specifically, we proceed in thefollowing way: (i) specify how to measure the macroeconomic growth; (ii)decompose the growth rate into contributions from each industry; and (iii)
clarify how structural changes, especially the emergence of new industries,
aﬀect the growth rate. From these considerations, it will be clear why thegrowth rate has cycles.
In our model, we can measure the macroeconomic growth rate in two
ways. The first metric is the conventionally measured per capita GDP
growth, denoted by gt . This metric measures how much the total value of output increases for fixed prices. Though it is relatively easy to observe its value
in the real economy, there is a shortcoming, as this metric does not accountfor the benefit resulting from increases in the variety of goods. Alternatively,
we can measure growth by the instantaneous utility, which incorporates thebenefit from a large variety of goods. However, this alternative also has its
intrinsic drawback, as instantaneous utility is not a cardinal metric, and thus
is not generally suited to measure the absolute value of welfare improvement.
Thus, there seems to be an inevitable tradeoﬀ between these two concepts.
As long as growth is confined to the equilibrium path, however, this is not thecase; we show in the following that there is a linear relationship between thetwo metrics, and thus we can in eﬀect use either of them without arbitrarinessof the metric and without missing certain benefits.
For this purpose, let us derive a formula for the growth rate of the instantaneous utility, U̇t /Ut . From the equation (12), which states that the
instantaneous utility is the reciprocal of the average price index, the growthrate can also be seen as the rate of fall of this index. Since the latter depends on the distribution of firms nt (·) and the profile of prices pt (·), we
must, in principle, deal with the time variation of these functions. For easeof exposition, however, let us again in this subsection utilize the discrete− 37 −

approximation. That is, we assume that firms are distributed exclusively
on grids with countable points, {s1 , s2 , · · · }.33 Then, a fundamental rule ofcalculus enables us to express the growth rate as34
¶
¶
∞ µ
∞ µ
X
U̇t X
−1 ∂ P̄t−1 ∂ P̄t=P̄t−P̄tṅtj .
(40)(−ṗt (sj )) +Ut∂p(sj )
∂nj
j=1
j=1
The first term in the above expression represents the welfare improvementthat comes from the decline in the price of goods, whereas the second term
represents the benefit from the increase in the variety of goods.
Applying the equilibrium condition (E100 ) for (40), the first term in the
RHS simply becomes gt , the per capita GDP growth rate, and the secondturns out to be (λ + gt )/(σ − 1). Since at each instant the equilibrium
allocation maximizes instantaneous utility subject to the resource constraint,
these results can be interpreted using the Envelope Theorem.35 Collecting
33

See Subsection 3.2.

34

Strictly speaking, we must show that the time derivatives of pt (sj ) and ntj actually exist. First, from the equilibrium conditions and (5), we have ṗt (sj ) = ȧt (sj ) =−Kt (sj )−2 K̇t (sj ). Since K̇t (sj ) is defined by (6), ṗt (sj ) is also well defined. Next, asshown in Subsection 3.3, the equilibrium firm distribution is the solution to the constrained minimization problem of P̄t . Since the constraint of this problem is compact andchanges continuously, we can apply the Theorem of the Maximum to show that the solution {nt1 , nt2 , . . . } changes continuously and thus their time derivatives exist. See Stokyand Lucas (1988, Chapter 3).
35

Recall that the per capita nominal income is normalized to unity and that the instantaneous utility function is homogeneous of degree one in consumption density. Thus,
given the number of firms, a decline in prices induces an increase in the per capita total
consumption quantities, which results in a proportional increase in the instantaneous utility. Therefore, the first term, the benefit from falling prices, exactly equals the per capita
GDP growth rate. Next, let us focus on the benefit from increasing variety, represented
by the second term. It will be shown in Section 6 that population growth aﬀects theinstantaneous utility by multiplying the number of firms with every characteristic, andthat the elasticity of instantaneous utility to population is (σ − 1)−1 . Thus, the first partof the second term, λ/(σ − 1), comes from exogenous population growth. The remaining
part, gt /(σ − 1), reflects a mixture of the above two eﬀects. That is, an increase in the
total quantity because of the decline in prices is absorbed, not by an increase in the output− 38 −

these eﬀects together, we now have a simple formula:
U̇t
σ
λ
=
gt +
.
Ut
σ−1
σ−1

(41)

The above states that, as far as the equilibrium path is concerned, there isa linear relationship between the two metrics of the macroeconomic growthrate. Thus, it would be a matter of convenience which of the two concepts
is used, and, in the following, we concentrate on the per capita GDP growthrate.36
In our model, the per capita GDP growth is driven by the learningby-doing process in each industry through the reduction of the unit costof producing a good with each characteristic. Then, how much does eachindustry contribute to the macroeconomic growth? To clarify the issue, we
derive an intuitive decomposition of gt . Formally, the definition of the percapita GDP growth isP∞dṄt
j=1 pt (sj ) dt (Qt (sj )ntj )gt ≡ P∞− ,(42)
Nt
j=1 pt (sj )Qt (sj )ntj
where Qt (sj )ntj represents the total amount produced in the industry atcharacteristic sj .37 The above expression simplifies considerably under theequilibrium conditions (E1)-(E4), but we can go one step further. DiﬀerenP
tiating both sides of the total expenditure relationship, ∞j=1 at (sj )ntj = Nt ,quantity of each firm, but by an increase in the number of firms, which induces an additional benefit from the larger variety. For a formal treatment of the Envelope Theorem,
see Mas-Colell, Whinston and Green (1995, Section M.L).
36

Note that, however, the relationship (41) holds only if the equilibrium conditions(E100 )-(E4) are satisfied. For example, industrial policies that accelerate gt do not necessarily improve the growth rate of the instantaneous utility, since under such policies theequilibrium conditions might not be satisfied.
37

To simplify the exposition, we proceed in the following text as if the support of every
industry contains no more than one point in S. However, more generally, the support ofan industry might be a finite number of adjacent points in S. In such a case, we shallinterpret this industry’s contribution to the growth rate as the sum of the contributions
from the firms at all points within its support.

− 39 −

we have an identity,
Nt−1

∞
X
j=1

at (sj )ṅtj = λ − Nt−1∞Xȧt (sj )ntj .

(43)

j=1

This formula enables us to express the growth rate in terms of the rate of cost
reduction, −ȧt (sj )/at (sj ), which is directly related to the learning-by-doing
process. Let us denote the latter by ηtj . Then, the equation (42) reduces to
gt =

∞
X

xtj ηtj ,

(44)

j=1

where xtj is the share of GDP produced by each industry, at (sj )ntj /Nt . The
above formula gives a clear decomposition of the per capita GDP growthrate into each industry; that is, gt is the weighted sum of the rates of cost
reduction in each industry, where the weights are the shares in the GDP.
While this formula might at first blush suggest that there is a linear
relationship between each industry’s share and its contribution to the growthrate, this is not the case, since the rate of cost reduction, ηtj , also depends
on shares through the learning-by-doing process. Specifically, from (5) and(6), we can write down the rate of cost reduction in terms of current shares,
unit costs and population:at (sj ) − 1(at (sj ) − 1)2 X xtm −ν|sj −sm |e−δ.
ηtj = γNtat (sj )a (s )at (sj )m=1 t m∞(45)

Each item within the summation in the first term represents the flow ofknowledge that spills from the industry at sm to the industry at sj . Notethat this item depends negatively on the distance between the two industries
and linearly on the industry’s share where production experience is gained.
Thus, we can see that the rate of cost reduction itself also depends linearlyon each industry’s share, given Kt (·) and Nt . Combining the two formula(44) and (45), it turns out that, at each instant, the per capita GDP growthdepends not linearly but, in fact, quadratically on shares in the GDP.

− 40 −

Figure 4: The per capita GDP growth and its decomposition
Numerical Analyses
Now, let us present a simulation result using the same parameter values andinitial conditions as in the previous section. Figure 4 depicts the overall
per capita GDP growth rate and its decomposition into contributions fromeach industry, according to formula (44). We can easily see from this figure
that, at each point in time, usually only one or two, at most three, young
industries represent quite a large portion of the growth rate. This observation is consistent with our formulation of the learning-by-doing process; thatis, since there is a lower bound on the unit cost of production, structuralchanges, especially the emergence of new industries, are necessary to sustain
positive economic growth. When we take a closer look at this figure, however, a paradoxical phenomenon can be observed: the overall growth rate
falls sharply immediately after the emergence or disappearance of industries,
and it is not until newly emerged industries become suﬃciently large thatthe growth rate seems to rebound. In the following, we attempt to explain
why structural changes aﬀect the growth rate in such a way, and eventually− 41 −

give rise to cyclical fluctuations in growth.
When a new frontier industry emerges, some portion of consumer expenditure begins to shift from existing industries to the new one. In our model,
where knowledge accumulation cannot be separated from production, thisevent has two distinct eﬀects on the growth rate. To clarify each of them, let
us diﬀerentiate the formula (44) with respect to each industry’s share:38
∞
X
∂gt
∂ηtm
= ηtj +
xtm
.
∂xtj
∂x
tj
m=1

(46)

The first term, ηtj , represents the eﬀect that comes from variation in thecomposition of GDP: if a larger proportion of GDP is produced by industries
in which costs are rapidly falling, the overall GDP growth rate is higher.
Figure 5 gives the rates of cost reduction in each industry, and the gap
between any two curves indicates the magnitude of the ‘GDP compositioneﬀect’ when a marginal share shifts between these industries. Apparently,
the rate of cost reduction is higher in young, recently emerged industries,
since their costs are still distant from the lower bound and thus are more
eﬀectively reduced by a given amount of production experience. Therefore,
when a marginal share shifts to new industries, the resulting variation in the
GDP composition has a positive eﬀect on the overall growth.
There is another eﬀect, however, which may overturn the above result.
Specifically, the second term in (46) comes from variations in the compositionof the learning-by-doing process: variations in each industry’s share aﬀect thepattern of knowledge accumulation and thus the pattern of cost reductions,which in turn has an impact on the overall growth rate. Diﬀerentiating (45),
we can write this ‘learning composition eﬀect’ as
γNt

∞
X

m=1

xtm

(at (sm ) − 1)2 1
e−ν|sj −sm | .
at (sm )at (sj )38(47)

The formulas derived earlier, (44) and (45), show that the per capita GDP growthdepends on current shares, unit costs and population. Since unit costs are determined by
the accumulated knowledge, however, they do not aﬀect the growth rate at each instant in
the first order. Also, population is exogenously determined. Thus, each industry’s shareis the only variable that influences the growth rate in the short term.

− 42 −

Figure 5: The GDP composition eﬀect
Although a marginal increase in the production experience in an industry
sj leads to cost reductions in every industry through spillovers, the magnitudeof the spillover, represented by the term e−ν|sj −sm | , diminishes as the characteristic distance gets larger. In fact, the major portion of the contribution
comes from the cost reduction in industry sj itself. Thus, a kind of economy
of scale exists: additional experience in a certain industry contributes greatly
to the growth rate when its own share is already large, because additional
cost reduction has a strong eﬀect on a large portion of the GDP.39 Figure 6
plots the value of the expression (47) for each industry, and again, the gap
between any two curves indicates the magnitude of the learning compositioneﬀect. There seems to be a typical pattern: each curve starts from a relatively low value, since a recently emerged industry captures only a small
share; then it grows until it attains its peak value around the time anothernew industry emerges. It then gradually declines as the unit cost approaches
39

Roughly speaking, when a small industry’s share shifts to a large one, the growth rate
tends to be enhanced and vice versa.

− 43 −

Figure 6: The learning composition eﬀect
the lower bound.
With these two results in hand, we can now explain the cyclical behavior of the growth rate. Immediately after emergence of a new industry, theindustry previously on the frontier still gets the largest share.40 Thus, theoverall growth rate is greatly aﬀected by the slowdown in its pace of learningby-doing when its share shifts to the small new industry. In fact, we can see
from the last two figures that the learning composition eﬀect is negative and,in absolute value, dominates the GDP composition eﬀect. This is the reason
why the growth rate falls following the emergence of a new industry, at leastin the short term.41 As the knowledge at the new industry accumulates,40

See Figure 3 in the previous section.

41

In a parallel way to the text, we can show why the growth rate tends to fall after thedisappearance of existing industries. Let us consider what happens just before an industry
disappears. Since its unit cost must have approached the lower bound, the rate of cost
reduction in this industry is generally small. Moreover, since this industry is going todisappear in the near future, it must have only a small share. In such a case, a marginal
share shift from it to other industries will enhance the growth rate, since both the GDP− 44 −

however, its share becomes gradually larger, whereas that of the previous
one becomes smaller. This implies that the absolute value of the learningcomposition eﬀect becomes small. Eventually, this negative eﬀect is cancelled by the GDP composition eﬀect, and thereafter the growth rate begins
to rebound. Furthermore, when the new industry’s share dominates that ofthe previous one, the learning composition eﬀect itself turns positive, accelerating the growth rate. However, this boom period does not last forever.
Another new industry will emerge, which starts another U-shaped pattern inthe growth rate.

6

Population Growth and Scale Eﬀect

Prior to this section, we have assumed that the size of the population grows
exogenously, which was a necessary condition for positive long-run growthin our model. This assumption is, however, not essential. We show herethat this assumption can be dispensed with by reformulating the process ofaccumulation of knowledge.
Recently, much attention has been paid to the relationship between population growth and long-run growth rate. In his influential paper, Jones (1995)
has pointed out that most endogenous growth models have an implausible‘scale eﬀect’: in these models the size of population aﬀects the long-run
growth rate. This phenomenon comes from a particular specification of the
R&D process, in which the current eﬃciency of R&D depends linearly on
the accumulated past experience of R&D. Jones (1995) presented an alternative specification, where eﬃciency does not depend (or depends less thanlinearly) on the past experience. However, this involves a tradeoﬀ: the scaleeﬀect is eliminated, at the cost of requiring exponential population growth tomaintain long-run growth. Although there is a diﬀerence between R&D andthe learning-by-doing process, our model essentially belongs to this alternacomposition eﬀect and the learning composition eﬀect are positive. After this industry
completely diminishes, however, such enhancing eﬀects also disappear, which tends tolower the growth rate.

− 45 −

tive class of specifications. Is this a flaw? Since it is controversial whether
the scale eﬀect is present or not, we do not argue this point in this paper.
Rather, we simply illustrate how this model can be extended for the casewhere population growth is not necessary, but the scale eﬀect is present.42
To this end, let us clarify how the population size aﬀects the instantaneous equilibrium in the original model. Observe that the unit demand price¡¢function, defined by (19), is homogeneous of degree zero in n(·), N . Thus,
if we double both the total population and the equilibrium distribution offirms simultaneously, they would continue to constitute an equilibrium, given
the distribution of knowledge. While such an event does not alter the priceof each good, the consumption density of each good must have been halved,
since now there are twice the varieties of goods, whereas the per capita income is unaﬀected. As a result, the utility of each consumer mildly increases
with the returns to scale being (σ − 1)−1 .43 This kind of scale merit exists
in almost all models that incorporate Dixit-Stiglitz style product diﬀerentiation. Rather, the major finding here is the fact that, given the distributionof knowledge, the size of the economy does not alter the basic structure ofindustries, but merely scales it.
From the viewpoint of economic growth, this fact implies that a larger
population simply accelerates the learning-by-doing process, and hence therate of economic growth at each instant.44 Indeed, the equation (45) showsthat the true coeﬃcient that determines the speed of the economy-wide
42

Some recent ‘hybrid’ models overcome this tradeoﬀ by combining the quality laddermodel with the variety expansion model. These models eliminate the scale eﬀect while,
nonetheless, long-run growth is possible with a constant population. In principle, it ispossible to achieve the same eﬀect in our model by adding one more dimension of variety
expansion to it. See Jones (1999) for a survey.
We see that the utility function, defined by (1) and (20 ), is homogeneous of degree
σ/(σ − 1) with respect to the distribution of firms, and of degree one in terms of theconsumption density. Therefore, if the population is doubled, the utility will be multipliedby 2σ/(σ−1) because of the doubling of firms, but simultaneously divided by two becauseof the halving of the consumption density.
43

44

Note that this fact does not mean that the size of the population aﬀects the ‘long-run’economic growth rate.

− 46 −

learning-by-doing process is not γ itself, but γNt . If this coeﬃcient were
fixed, the long-run growth rate would eventually fall to zero, since the frontier industries’ share would get smaller and smaller compared to the growing
whole economy. In our original model where Nt grows exogenously, the true
coeﬃcient γNt also grows linearly with the size of the economy, enabling
positive long-run growth. Alternatively, suppose that the total population
is fixed, but the eﬃciency of learning-by-doing, γ, grows in accordance withpast experience of learning-by-doing, namely, the whole profile of accumulated knowledge. For example, this consideration can be specified asZ ∞Γ(Kt (s)) ds,
γt =−∞where Γ(·) is an increasing function. This specification has virtually the sameeﬀect as population growth and resolves the zero growth problem for a fixed
population, with the cost that this model now exhibits the scale eﬀect.

AppendixA

Proof of Concavity of Ũ

Here we establish concavity of the transformed instantaneous utility functionŨ, given by (10 ) and (200 ), in terms of {nj } and {hj (·)}.
Consider two sets of variables, θ = {{n0j }, {h0j (·)}} and θ0 = {{n00j }, {h00j (·)}}.
In addition, for arbitrary α ∈ (0, 1), let us construct a new set of variablesθ∗ = {{n∗j }, {h∗j (·)}} by settingn∗j = (1 − α)n0j + αn00jh∗j (·) = (1 − α)h0j (·) + αh00j (·)

for j = 1, 2, · · · .

(A.2)

Our task is then to show that Ũ ∗ ≥ (1 − α)Ũ 0 + αŨ 00 holds, where Ũ ∗ is thevalue of the transformed utility function when we have substituted for θ∗ ,and so on.
First, from an identity 1/σ + (σ − 1)/σ = 1, we have
1

σ−1

1

σ−1

1

σ−1

n∗j σ h∗j (r) σ ≥ (1 − α)n0j σ h0j (r) σ + αn00j σ h00j (r) σ ,− 47 −(A.3)

for all j and r. Applying this result for (200 ), it is straightforward to showσ−1v ∗ (r) σ =

∞
X

σ−1

1

σ−1

n∗j σ h∗j (r) σ e−τ σ |r−sj |

(A.4)

j=1

≥ (1 − α)v 0 (r)σ−1σ+ αv 00 (r)

σ−1
σ

,

where v ∗ (r), v 0 (r) and v 00 (r) are defined in parallel with Ũ ∗ , and so on. Next,
note that from the assumption σ > β > 1, an inequality σ/(1−σ)·(β−1)/β <1 holds. Thus, the previous result can be transformed into
∗

v (r)

β−1
β

h
i σ β−1
σ−1
σ−1 1−σ β000σσ≥ (1 − α)v (r)+ αv (r)0≥ (1 − α)v (r)

β−1
β

00

+ αv (r)

β−1
β

(A.5)

.

Finally, applying the above inequality for (10 ), we have
Z ∞
β−1
∗
Ũ =
v ∗ (r) β dr ≥ (1 − α)Ũ 0 + αŨ 00 ,

(A.6)

−∞

which establishes the claim.

B

Proof of Uniqueness of Optimum

In the text, we have established that the set of conditions (K1)-(K3) is thenecessary and suﬃcient set of conditions for the instantaneous optimum, andthat the set of maximizers is convex. Here we prove that there is no morethan one maximizer.
Suppose that there are two sets of variables, θ0 and θ 00 , defined the same
way as in Appendix A, and that both of them attain the maximum transformed utility, denoted by ŨM . When we choose α ∈ (0, 1) arbitrarily, another set of variables θ∗ , defined by (A.2), will also attain ŨM , since the setof maximizers is convex. This means that the inequality (A.6) holds withequality. By making use of this fact, in the following we show that θ0 = θ00
necessarily holds, which establishes the uniqueness of the maximizer.
For the final inequality (A.6) to hold with equality, the second inequalityin (A.5) must hold with equality for all r ∈ R. This in turn requires v 0 (r) =− 48 −

v 00 (r) for all r ∈ R. On the other hand, since both θ0 and θ00 are maximizers,
the condition (K2) is satisfied by both of them. Substituting this condition
for (200 ), we have
0

v (r)

σ−1
β

=

µ

σ β−1 1
σ − 1 β Nξ 0

¶σ−1 X∞n0j a(sj )e−τ (σ−1)|r−sj | ,

(A.7)

j=0

and a similar relationship for the case of θ00 . Note that if we see each termin the summation, a(sj )e−τ (σ−1)|r−sj | , as a function of r, they are linearly
independent. Thus, for the equation v 0 (r) = v 00 (r) to hold for all r, allcoeﬃcients must be equal to each other: that is, ξ 0 1−σ n0j = ξ 00 1−σ n00j holds forall j. This implies that there is a positive constant ζ, such that n00j = ζn0j
holds for all j.
Next, let us turn to the former two inequalities. For the final inequality tohold with equality, the first inequality in (A.5) must also hold with equality
for all r. This is equivalent to (A.4) holding with equality, which in turnrequires (A.3) to hold with equality for all r and j. Note that from theargument above, if n0j = 0 for some j, then n00j = 0 also holds. In this case,condition (K2) ensures that h0j (r) = h00j (r) = 0 for all r. If n0j > 0, on theother hand, then n00j = ζn0j > 0 holds and condition (K2) ensures that both
h0j (r) and h00j (r) are positive for all r. In such a case, for (A.3) to hold withequality, an equality h0j (r)/n0j = h00j (r)/n00j must hold for all r. After all,h00j (r) = ζh0j (r) is required for all j and r.
Finally, remember that both θ 0 and θ00 satisfy the resource constraint(K3). The preceding results, n00j = ζn0j and h00j (r) = ζh0j (r) for all j and r,
are compatible with this condition if and only if ζ = 1. This establishes theclaim.

C

Upper and Lower Bounds for Ψ(s)Here we establish upper and lower bounds for the function Ψ(·), defined by(36). Specifically, we prove that an inequality 1 ≤ Ψ(s) ≤ eν|s| holds, withequality if and only if s = 0.

− 49 −

Let us define a new function ψ(·, ·) by
ψ(r, s) = (σ − β)τ |r| − (σ − 1)τ |r − s|.

(A.8)

Then, we can express the function concerned, Ψ(·), as
Ψ(s) ="R ∞exp ψ(r, s) drR−∞∞
exp ψ(r, 0) dr
−∞

# σ1

.

(A.9)

With this expression, it is obvious that Ψ(0) = 1 holds, since the numerator
and the denominator coincide when s = 0.
Next, consider the case of s > 0. We can show thatψ(r, s) ≤ ψ(r − s, 0) + (σ − β)τ s,(A.10)

holds with strict inequality when r < s. Substituting this result into (A.9),
we haveσ−βΨ(s) < e σ τ s .

(A.11)

Recall the assumptions σ > β > 1 and ν > ((σ − 1)/σ)τ , which together
establish an inequality ν > ((σ − β)/σ)τ . Applying this for (A.11), theinequality Ψ(s) < eνs results. Similarly, we can also show that
µ
¶
σ−1
s, 0
(A.12)
ψ(r, s) ≥ ψ r −β −1
holds with strict inequality if 0 < r < ((σ − 1)/(β − 1))s. Substituting
this into (A.9), we have Ψ(s) > 1. Collecting both results, the inequality
1 < Ψ(s) < eν|s| is established for all s > 0.
Finally, since Ψ(s) is symmetric around zero, we can show the same result
for s < 0.

D

Existence of the Critical Value K̄ ∗Here we prove that there is a finite positive constant K̄ ∗ such that the market
potential function for the one-industry economy (37) is negative for all s > 0
if and only if K̄0 ≤ K̄ ∗ .
− 50 −

The negativity of (37) turns out to be equivalent toK̄0 ≤eνs − Ψ(s)≡ κ(s)
Ψ(s) − 1(A.13)

for all s > 0. Then, if we can show that the minimum of κ(s) exists in theinterval (0, ∞), the minimized value is K̄ ∗ and we are done.
Note that, since 0 < Ψ(s) < eνs holds for all s ∈ (0, ∞), the function κ(s)is well defined and strictly positive in this interval. Moreover, the continuityof Ψ(s) ensures the continuity of κ(s). The interval (0, ∞) is, however, notcompact and thus we must examine how κ(s) behaves at both ends of theinterval.
First, consider the case when s tends to zero from above. In this case, both
the denominator and numerator of κ(s) also tend to zero. Then, applying
the l’Hôpital’s Theorem, we haveν − Ψ0 (s).
s→+0Ψ0 (s)lim κ(s) = lim

s→+0(A.14)

Note that from (A.8) and (A.9), we can confirm that Ψ(s) is increasing forall s > 0. In addition, since Ψ(s) is everywhere diﬀerentiable and symmetricaround zero, Ψ0 (0) = 0 apparently holds. Thus, as s tends to zero from above,Ψ0 (s) also tends to zero from above. Substituting this result for (A.14), we
have lims→+0 κ(s) = +∞.
Next, let us examine how κ(s) behaves as s tends to infinity. From theinequality (A.11) and the assumption that parameter ν is greater than ((σ −β)/σ)τ , we can show that e−νs Ψ(s) tends to zero from above as s tends toinfinity. Similar arguments can be given for e−νs (Ψ(s) − 1), since Ψ(s) > 1
holds for all s > 0. Thus, we have
1 − e−νs Ψ(s)= +∞.
s→∞ e−νs (Ψ(s) − 1)lim κ(s) = lims→∞(A.15)

These arguments assure that the minimizer of κ(s) exists in the interiorof (0, ∞), and its minimized value, K̄ ∗ , is finite and strictly positive.

− 51 −

Acknowledgements
We are grateful to Kenn Ariga, Masahisa Fujita, Koichi Futagami, and Akihisa Shibata for their helpful comments and suggestions. The author is
responsible for any remaining errors.

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