Discussion Papers In Economics
And Business

Wants and Past Knowledge:
Growth Cycles with Emerging Industries
Ryo Horii

Discussion Paper 06-03

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWants and Past Knowledge:
Growth Cycles with Emerging Industries
Ryo Horii

Discussion Paper 06-03

January 2006

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANWants and Past Knowledge:
Growth Cycles with Emerging Industries∗
Ryo Horii† (Osaka University)
January 9, 2006

Abstract
This paper develops a theory of endogenous growth cycles focusing on theinteraction between consumers’ desire to satisfy an indefinite range of wants
and firms’ incentive to utilize knowledge from past production experiences. We
show that firms endogenously form a number of distinguishable industries asaccumulated knowledge induces them to agglomerate in the technology space.
Knowledge accumulation in existing industries reduces production costs, but, as
the diminishing returns from learning sets in, some firms start to adopt previously
unexplored technologies so that their new goods fit consumers’ unsatisfied wants
and attract large demand. Thus, sporadic emergence of new industries generates
growth cycles, where both the timing and the new technology to be adopted
are endogenously determined. New industries based on new technology reduce
the rate of per capita GDP growth in the initial phase, but nonetheless areindispensable for sustained economic growth in the long run.
JEL Classification Numbers: O31, O33, O41.
Keywords: endogenous growth, technology choice, wants, knowledge, cycles.

∗

This research is partially supported by the Japan Society for the Promotion of Science, Grant-inAid for Young Scientists (B), 2004—2006. I am grateful to Kenn Ariga, Elias Dinopoulos, Masahisa
Fujita, Koichi Futagami, Masayuki Ohtaki, Mark Sanders, Paul Segerstrom, Akihisa Shibata, Katsuya
Takii, Klaus Wälde, and seminar participants at the Econometric Society World Congress 2005,
EEA Amsterdam 2005, ISER, Kobe University, Kyoto University, Tokyo University, and Yokohama
City University for their valuable comments and suggestions on earlier versions of this paper. Anyremaining errors are naturally mine.
†

Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka 560-0043,
JAPAN. Tel: +81-6-6850-5254, Fax: +81-6-6850-5274, E-mail: horii@econ.osaka-u.ac.jp1Wants and Past Knowledge January 9, 2006

1

2

Introduction

The notion that long-term economic growth is primarily the result of the growth oftechnological knowledge is now a widely held view among growth economists. Not entirely surprisingly, this perception existed in the 1960s when Schmookler published his
influential volume entitled “Invention and Economic Growth,” in which new additions
to knowledge were termed invention or subinvention according to the degree of novelty.
The focus of the book was on the determinants of production of new knowledge, stilla central concern of the current growth literature. He wrote as follows.
The very definition of an invention as a novel combination of pre-existingknowledge to satisfy some want better suggests the possible causes for its
occurrence. ... Since it is based on prior knowledge, the received stock ofknowledge must also play a role. And since it is calculated to better serve
human wants, these too must also aﬀect invention. ... [O]ur chief inquiry
concerned the comparative influence of wants and past knowledge on theinventive process. (Schmookler 1966, p. 197)
Given that there are broad classes of technological knowledge, his insight suggests
two mechanisms by which past economic activities aﬀect the pattern of knowledge accumulation today. The stock of knowledge upon which a new firm can rely is often
obtained through the past experiences of existing firms. Thus, a new firm has an incentive to adopt the same technology as existing firms to utilize that class of knowledge,
which again gives subsequent firms the same, and further fortified, incentive.1 This
force gives rise to a group of firms–which we call an industry in this paper–that utilize the common knowledge base of a particular technology to minimize their operating
1

Jovanovic and Nyarko (1996) and Redding (2002) examined this issue and showed that techno-

logical lock-in may occur if agents accumulated too much knowledge with respect to a particulartechnology. In our model, technological lock-in does not persist because of a conflicting force thateventually dominates; namely, human wants.

Wants and Past Knowledge January 9, 2006

3

costs.
Human wants aﬀect the decisions of firms in a diﬀerent way. Any technology
is subject to an obvious limitation regarding the range of wants to which it can beadequately applied, and this fact influences a firm’s technology choice. While this
aspect of technology is not considered explicitly by Schmookler, its importance wasafterward pointed out by Rosenberg (1974).
Oddly enough then, science and technology play a subordinate role in influencing the direction of inventive activity within Schmookler’s analysis,
not because his analysis downgrades their historical significance, but rather
because he regards science and technology in the modern age as being, ina significant sense, omnicompetent. ... Now this is precisely the aspectof Schmookler’s argument which seems to be most inadequate. ... Many
important categories of human wants have long gone either unsatisfied orvery badly catered for in spite of a well-established demand. (pp. 94, 97)
Given the indefinite range of human wants, Rosenberg’s view implies that there will
always be a set of wants that are not well matched by existing technologies. Firms
can attract large demand if they can adopt previously unexplored technologies so that
their new goods fit consumers’ unsatisfied wants. Thus, profit maximizing firms may
find it optimal to adopt technologies significantly diﬀerent from other firms, rather
than to exploit the benefits of accumulated knowledge within existing industries.
There is an apparent conflict between knowledge and wants–or cost and demand–in the choice of technology by firms, and therefore in the direction in which knowledge
grows.2 This paper develops a theoretical framework that explicitly captures this trade2

Empirically, it has been diﬃcult to distinguish between these two forces because the strength ofboth forces depends on closeness between firms. Nonetheless, Bloom, Schankerman and Van Reenen(2005) recently developed two measures of a firm’s position in technology space and product market
space and found that both technology spillovers (i.e., past knowledge) and product market rivalry(i.e., human wants) aﬀect a firm’s value.

Wants and Past Knowledge January 9, 2006

4oﬀ and investigates the pattern of economic growth, based on variety-expansion models of endogenous growth (e.g., Romer 1990; Grossman and Helpman 1989). We showthat firms endogenously form a number of distinguishable industries, as accumulated
knowledge induces firms to agglomerate in technology space. Knowledge accumulationin existing industries reduces production costs, but, as the diminishing returns fromlearning set in, some firms start to adopt new and significantly diﬀerent technologies
to capture demands for unsatisfied wants. Both the timing and the new technology to
be adopted are endogenously determined in equilibrium. As a result, the equilibrium
dynamics are characterized by the sporadic emergence of new industries, rather thana smooth increase in the number of symmetric products.
This type of dynamics naturally causes the rate of economic growth to fluctuate.
In particular, it captures the observed tendency that the emergence of a new industry
that utilizes a new technology–e.g., electricity and information technology–reduces
the rate of per capita GDP growth in the initial phase, a phenomenon known as
the “productivity slowdown puzzle.”3 It is shown that this slowdown occurs because
emerging industries diversify the GDP share of individual industries and diminish thebenefits of the agglomeration economy that comes from knowledge accumulation within
an industry.
The model also shows that new industries nonetheless disproportionately contributeto the economy’s productivity growth after becoming large.4 As knowledge concerning
the new technology accumulates, more firms enter the new industry, further accelerat3

Hornstein and Krusell (1996) and Greenwood and Yorukoglu (1997) found a negative eﬀect of

the arrival of information technology (IT) on productivity, and attributed the productivity slowdown
to the learning cost associated with new technologies. Jovanovic and Rousseau (2005) found that a
productivity-growth slowdown occurred not only in the initial phase of the IT era, but also in theinitial phase of electrification.
4

In the case of the revivals in total factor productivity (TFP) growth during the period 1995—99,

Gordon (2000) argued that productivity increased only within the computer industry and associatedsectors, which together comprise only about 12 percent of the private business economy.

Wants and Past Knowledge January 9, 2006

5

ing knowledge accumulation. Production costs and prices of products fall faster, andeventually cause a spurt in overall GDP growth. It is shown that, at every point in
time, only a small number of young industries account for quite a large proportion of
the overall growth rate in the model. Without the sporadic emergence of new industries
induced by human wants, cost reductions within existing industries would eventually
come to an end.
As a study of cyclical growth, the present paper is closely related to Helpman andTrajtenberg (1998). They considered an exogenous process in which new general purpose technologies (GPTs) arrive sporadically and discretely, and showed that GPTs
trigger recurrent cycles in the growth rate. Our study improves the understanding of
this type of dynamics in that both the timing of arrival (emergence of new industries)
and the degree of discreteness (the diﬀerence between the technologies) are endogenized.
This paper is also related to the literature that integrates endogenous fluctuation,
sometimes called “natural volatility” as opposed to exogenous shocks, with endogenous
long-run growth. Freeman, Hong and Peled (1999), Wälde (2002, 2005), Maliar andMaliar (2004), and Phillips and Wrase (2005) developed models in which activity ofagents causes major improvements in the TFP to occur sporadically, thereby generating
fluctuations in the rate of economic growth. A common feature of these studies is thatthey assume that productivity improvements in each sector are discrete, and that the
number of sectors are finite so that the law of large numbers does not work. In contrast,
we model a continuous-time economy with a near continuum of technologies, whose
productivity improves always continuously. Growth cycles occur in our model because
spillover of knowledge generates a centripetal force that induces firms to choose only afinite number of technologies, and consumers’ wants generate a centrifugal force that
expands the set of technologies used in discrete steps.
The rest of this paper is organized as follows. Sections 2 and 3 respectively present
the model of human wants and that of past knowledge. Section 4 derives the equi-Wants and Past Knowledge January 9, 2006

6librium distribution of firms at each instant, given the distribution of knowledge accumulated by that time. Section 5 investigates how the interaction between wants
and past knowledge drives the evolution of firms, and hence determines the pattern ofknowledge accumulation. Section 6 examines the fluctuations in growth rate. Section
7 concludes.

2

Modeling Human Wants

Preferences
In the model, the economy consists of a continuum of identical consumers with measure
Lt , which grows at an exogenous rate λ > 0. They have an unbounded range ofwants, represented by real number r ∈ (−∞, ∞). Each want can be satisfied byconsuming diﬀerentiated goods and, as in Dixit and Stiglitz (1977), consumers benefit
from consuming a large variety of diﬀerentiated goods to satisfy each want. In ourmodel, however, some goods are more suitable for satisfying a particular want thanothers, because goods are produced using diﬀerent technologies and therefore a great
variability exists in the wants that the goods best satisfy.
Let goods be indexed by i ∈ [0, It ], where It is the measure of the total numberof goods at time t, which is to be determined in equilibrium. Let set (i) ∈ (−∞, ∞)denote the technology by which good i is produced. Technologies are indexed sothat goods produced by technology set (i) are best suited for satisfying want r = set (i).

The consumer can also arbitrarily divide good i to use part of it to satisfy any wantr 6= set (i), but the good becomes gradually unfit as want r and technology set (i) diverge(See Figure 1).

The representative consumer decides the amount of good i to be consumed tosatisfy want r at each point in time. Let xt (r, i) denote this amount, or more precisely
consumption density defined for all wants r ∈ (−∞, ∞) and all goods i ∈ [0, It ]. Then

Wants and Past Knowledge January 9, 2006

7

Figure 1: Technologies and wants.

the level of satisfaction of want r is:·Z It¸σ/(σ−1)¡¢−τ |r−est (i)| (σ−1)/σxt (r, i)edi,vt (r) =(1)0

where τ > 0 is a coeﬃcient that measures how rapidly a good becomes unfit as its
technology diverges from r, and σ > 1 represents the elasticity of substitution between
goods. The consumer’s overall instantaneous utility at time t is given as an aggregation
over v(r):
ut =

·Z ∞

−∞

vt (r)(β−1)/βdr¸β/(β−1),(2)

where β ∈ (1, σ) is the elasticity of substitution between wants. The assumption of β <σ implies that substituting one want with another is more diﬃcult than substituting
one particular variety of good with another to satisfy the same want. Later we willshow that this property makes the consumer value novel goods that are made withtechnologies significantly diﬀerent from existing goods.
Each consumer inelastically supplies one unit of labor, and prices are normalized
so that the nominal wage at each instant is 1. Every good is perishable and cannot
be stored. In addition, as will be seen below, there is no opportunity to invest. Thus,
the homogeneity of consumers implies that the credit market involves no trade andevery consumer spends all income at each instant. Let pt (i) denote the price of goodWants and Past Knowledge January 9, 2006

8

i. The consumer then maximizes the instantaneous utility (2) with (1) under the
instantaneous budget constraint:
Z It µZ ∞0¶xt (r, i) dr pt (i) di = 1.

−∞

(3)

Consumer Demand
Here we derive the demand function of consumers for each good. Since the utility
function is a two-stage CES function, we adopt a two-step method. The first step is tomaximize the subutility of each want v(r) defined by (1) with respect to xt (r, i), underRI
the constraint 0 t pt (i)xt (r, i) di = yt (r). Here yt (r) is the density of expenditure forwant r, which in this step we take as given. For each want r and good i, the solution
to this problem and the maximized value are:xt (r, i) = yt (r)qt (r)σ−1 pt (i)−σ e−(σ−1)|r−est (i)| ,(4)

vt (r) = yt (r)/qt (r),·Z It¸−1/(σ−1)¡¢
τ |r−st (i)| −(σ−1)
where qt (r) ≡di.
pt (i)e

(5)
(6)

0

Function qt (r) represents the amount of expenditure required to increase one unitof vt (r), and therefore we call it the ‘price index’ of want r. Substituting the indirect
subutility function (5) for (2) gives the instantaneous utility in terms of the expendituredensity yt (·). The second step is to maximize this utility function subject to theR∞instantaneous budget constraint −∞ yt (r) dr = 1. The optimal expenditure density
and the maximized instantaneous utility, respectively, are:yt (r) = Qtβ−1 qt (r)−(β−1)
ut = 1/Qt ,·Z ∞¸−1/(β−1)−(β−1)
where Qt ≡qt (r)
dr
.

(7)
(8)
(9)

−∞

Qt is the expenditure function required to attain a unit instantaneous utility, which
can be interpreted as the ‘average price index’ over all wants. Substituting (7) into (4)Wants and Past Knowledge January 9, 2006

9yields the amount of good i used to satisfy want r as:
qt (r)σ−β pt (i)−σ e−(σ−1)τ |r−est (i)| .
x(r, i) = Qβ−1t(10)Integrating the demand for good i in (10) across all wants and then multiplying itby the population Lt yields the demand for good i:
pt (i)−σXt (i) = Lt Qβ−1tZ ∞qt (r)σ−β e−(σ−1)τ |r−est (i)| dr.

(11)

−∞

As shown by (11), the price elasticity of demand is constant. Therefore, the inverse
demand function for good i can be expressed as:
st (i))Xt (i)−1/σ ,pt (i) = dt (e(12)

where dt (s) is the unit demand price, which represents the price level at which any
good made with technology s sells a unit quantity. From (11), it is:
dt (s) ≡·Lt Qβ−1tZ ∞−∞qt (r)

σ−β −(σ−1)τ |r−s|

e

dr

¸1/σ

.

(13)

Centrifugal Force
Equation (13) implies that the price consumers are willing to pay for a variety of gooddepends on its technology s, and this dependence comes from variations in the priceindex qt (r) across wants.5 More specifically, from σ > β > 1, (13) shows that theunit demand price for a certain technology s is higher when the price index for thewants that are close to s is higher, and, from (6), this is the case when there arefewer goods produced by technologies that are close to s. This property has a naturalinterpretation: because of diminishing marginal utility from each want, consumers arewilling to pay more for a variety of good that fits wants that are not well met by theother goods. Figure 2 depicts an example of the shape of function dt (s), which we callthe unit demand price curve.

5

Observe that, if qt (r) is the same for all r, then d( s) is also the same for all s.

Wants and Past Knowledge January 9, 2006

10

Figure 2: Unit demand price curve. This example shows the shape of function dt (s) whengoods are produced with either of the three technologies (s, s0 or s00 ). The height of the gray bars
represents the number of varieties produced by each technology.

From the viewpoint of producers, which will be introduced shortly, they can sell
the same amount of their products at a higher price when they choose a technologythat is more distant in technology space from that of other firms since they will thenface little competition from producers who are using the same or similar technologies.
Thus, producers have an incentive to choose technologies that are as distant as possible
from others; creating a centrifugal force working in the technology space. This might
imply that the range of choices of technologies by profit-maximizing producers should
explode in the technology space. In reality, however, we do not observe such explosion
because, at a given date, the costs of producing goods extremely diﬀerent from others
are prohibitively high, if not impossible, due to scarcity of relevant knowledge.6 In thefollowing section, we describe the production side of the economy and how it is aﬀectedby the stock of knowledge that is accumulated in the past.

6

For example, even though a tour to Mars is quite diﬀerent from any other leisure services andit is not theoretically impossible, currently no business firm oﬀers such a tour, probably due to theextremely high estimated costs that stem from scarcity of knowledge or experience.

Wants and Past Knowledge January 9, 2006

3

11

Modeling Past Knowledge

Production Technologies
Each variety of goods is produced by a distinct firm. With a flow fixed cost that mustbe paid throughout the period of its operation (such as the cost of maintaining the
production line), any firm can enter the goods market.7 Firms can also exit the market
costlessly.
Each firm chooses one technology from a set of usable technologies denoted bys ⊂ (−∞, ∞). The number of potentially (or theoretically) usable technologies is verylarge and there are many analogous but slightly diﬀerent technologies in the set, eventhough only a small subset of s would be in use at any point in time. Specifically, defines by a countable set {sj }∞j=−∞ , where sj = jε with ε > 0 being a very small constant.
Set s does not change over time, and therefore it is defined as broadly as possible.8
Thus, our focus is on the evolution of knowledge about how to use technologies, while
abstracting from the growth of knowledge about what technologies exist.9
Goods are produced from labor with a constant marginal cost. Suppose that a
firm, call it firm i, produces an amount Xt (i) of its goods with technology set (i) ∈ s.

We normalize the units for the quantity of goods and for the measure of firms so that7

Most existing R&D-based growth models consider a lumpy setup cost (e.g., the cost of innovation)

while ignoring the flow fixed cost. Obviously, the reality incorporates both. The diﬀerence between
the two specifications is not so large as it might appear, because the flow fixed cost can also beinterpreted as the interest payments or dividends for the investors who financed the initial setup cost.
Nonetheless, the assumption that entry is reversible significantly simplifies the analysis since it enables
us to abstract from the forward-looking aspects of firms’ decisions.
8

At the limit to which ε → 0, s approaches the set of all real numbers, (−∞, ∞), which coincides

with the space of wants. The only reason that we maintain the assumption that s is countable is toprove the existence of the unique equilibrium in a tractable manner.
9

By the definition in Mokyr (2002), the former is called prescriptive knowledge while the latter is

propositional knowledge.

Wants and Past Knowledge January 9, 200612

the labor requirement for this activity is written by:10`t (i) =µσ−11
Xt (i) +σσ¶st (i)).
ct (e

(14)

In equation (14), the marginal and fixed flow costs of production in terms of eﬃciency
units of labor are normalized, respectively, to σ/(σ − 1) and 1/σ. The last term,st (i)), represents the number of workers required to generate one eﬃciency unitct (est (i)) also represents theof labor. Since the nominal wage is normalized to 1, ct (eexpenditure required for one eﬃciency unit of labor, and therefore we call it unit cost.
This cost depends on technology and is negatively related to the amount of knowledge
based upon that technology, as we now explain.
Learning by Doing
As people use a certain technology more, they learn how to use it better, and costs
associated with the use of that technology fall. Here we formalize this ‘learning-bydoing’ process.11
For each s ∈ s, let kt (s) denote the cumulative past experience of technologys. The value of kt (s) ≥ 0 can be interpreted as the amount of technology-specificknowledge on s. This paper considers aggregate or social knowledge in that it is10

Since our instantaneous utility function exhibits homogeneity of degree one in consumption den-sity, we can freely choose measurement units for output quantity. We can also normalize the number(measure) of firms: if we double the whole distribution of firms and simultaneously halve the output of each firm and the fixed requirement, the instantaneous utility will simply be multiplied by21/(σ−1) without substantially aﬀecting the behavior of agents. This normalization is similar to theone employed in Fujita, Krugman and Venables (1999, Chapter 4).
11

Of course, investment in knowledge often precedes production, for example in the case of R&D,

and many workers invest actively in their human capital. Although these processes 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 the adoption of existing technologies
than on the invention of new ones, with his rough estimate that in the U.S. adoption costs outweighinvention costs by a factor of 20 or 30 to 1.

Wants and Past Knowledge January 9, 2006

13

totally nonappropriable and every firm in the economy has access to it. For everys ∈ s, technology-specific knowledge grows according to:Z
Xt (i) di − δkt (s),k̇t (s) =(15)

set (i)=s

where the first term represents the aggregate amount of production using technologys and δ ≥ 0 is the rate of depreciation (or forgetting) of experience.
Past knowledge enables firms to produce goods with a fewer number of workers.
In this process, we focus on two important properties. The first property is spilloversof knowledge across diﬀerent technologies.12 Many of the technical and managerial
advances brought about by experience in the production of certain goods have applications elsewhere. That is, productivity increases using a particular technology are
not only a consequence of productive activity using that technology, but also the result of spillovers from learning-by-doing using other technologies. We assume that theamount of experience that a firm using technology s can rely on is proportional toP0−ν|s−s0 |kt (s0 ), where term e−ν|s−s | represents the extent to which the experiences0 ∈s ewith some technology can be applied to the production process with another technology.13 Parameter ν > (σ − 1)τ /σ represents how rapidly this applicability deteriorates
with the distance in the technology space.
Second, the learning-by-doing process eventually runs into diminishing returns.14
12

Young (1991, 1993) pointed out that there are substantial spillover eﬀects in the development ofknowledge across industries. See also Bloom, Schankerman and Van Reenen (2005)
13

Young (1993) focused onP0s0 ≤s kt (s ), implicitly assuming that the applicability of knowledge

does not depend on the distance between technologies. A more general specification is considered in
Stokey (1988).
14

For example, Gordon (2000, p.63) wrote: “Numerous industries have run into barriers to steady

growth in productivity, most notably the airline industry when jet aircraft reached natural barriersof size and speed, and the electric utility industry when turbogenerator/boiler sets reached naturalbarriers of temperature and pressure. The apparent dearth of productivity growth in the construction
and home maintenance industry reflects that electric portable power tools could only be invented onceand have been subject to only marginal improvements in recent decades.”Wants and Past Knowledge January 9, 2006

14

Figure 3: Unit cost curve. This example shows the shape of function ct (s) when past knowledge
is accumulated on three technologies (s, s0 or s00 )Specifically, intertemporal spillovers of experience generate a learning curve, in which
the unit cost starts high, decreases rapidly on initial units, and then begins to levelout. A simple specification we employ is:
Ã
!−1
X
0
e−ν|s−s | kt (s0 ).
ct (s) = 1 +

(16)

s0 ∈s

Equation (16) means that, while the unit cost of using technology s is a decreasingP
0function of usable knowledge s0 ∈s e−ν|s−s | kt (s0 ), the marginal contribution of knowl-edge becomes smaller when the production cost approaches the lower bound, which isnormalized to unity.

Figure 3 shows an example of the shape of function ct (s), which we call the unit cost
curve. Accumulated past production experiences of a particular technology reduce theunit cost of production by using the same technology. In addition, thanks to spillovers
across technologies, such experiences also contribute to lowering the cost of using othertechnologies, although to a lesser extent. Since this spillover eﬀect diminishes with the
distance in the technology space, past experiences enable firms to produce goods bysimilar (or ‘near’) technologies with lower labor costs. This mechanism generates acentripetal force that provides firms with an incentive to agglomerate with each otherWants and Past Knowledge January 9, 2006

15

over time.
Behavior of Firms
Now we are ready to examine firms’ profit maximization behavior. At each date t, afirm’s problem is twofold: it must choose technology set (i) and then price pt (i).

Let us start with the second stage: the choice of price given the choice of technology

set (i). Note that, since there is a continuum of firms, no single firm’s behavior willchange price indices qt (r) or Qt and thus firms take unit demand price (13) as given.

In addition, since knowledge is nonappropriable, they do not consider the impact of
their activities upon the accumulation of knowledge; i.e., they also take unit cost (16)
as given. Therefore, at each date, firm i chooses price pt (i) so as to maximize its current
profit π(i) = pt (i)Xt (i) − `t (i), subject to inverse demand function (12) and production
technology (14). The profit maximizing price and quantity, and the maximized profitare:st (i)),
pt (i) = ct (e(17)

Xt (i) = (dt (est (i))/ct (est (i)))σ ,(18)

πt (i) = σ −1 ct (est (i))1−σ (dt (est (i))σ − ct (est (i))σ ) .

(19)

Given those results, the problem in the first stage is straightforward: firms choose
the technology that maximizes the right-hand side of (19). Note that the amountst (i)). Thus, firms have anof profit, given by (19), is decreasing in the unit cost ct (eincentive to behave in a similar manner to other firms, since this would enable themto fully utilize the past stock of knowledge. However, since the amount of profit isst (i)), they also have an incentive to behave as
decreasing in the unit demand price dt (ediﬀerently as possible from others in order to capture consumer’s unsatisfied wants. In
the next section, we examine how these two forces uniquely determine the equilibrium
choice of technologies at each instant.

Wants and Past Knowledge January 9, 2006

4

16

Instantaneous Equilibrium

In this section, we characterize the market equilibrium at each instant, taking as given
the accumulated stock of past knowledge. Since the amount produced and the pricecharged by each firm is already obtained in (17), the main task of this section is toderive the equilibrium pattern of the choice of technology by firms. We then check theequilibrium of the labor market.
Recall that, in this economy, firms do not incur any cost when they enter or exit themarket. By the assumption of free entry, there must be no opportunity in equilibrium
to attain positive profit. Using (19), this condition is written as: dt (s) ≤ ct (s) for alls ∈ s. In addition, the profit of any firm must not be negative because they can exit
st (i)) = ct (est (i)) for all i ∈ [0, It ].
costlessly. That is, dt (eRecall that the unit cost function ct (s) is determined by past knowledge, which istaken as given at each instant. On the other hand, the unit demand price function
dt (s) is determined by the current pattern of the choices of technologies, or the current
distribution of firms on the technology space. Therefore, the search for the market
equilibrium is tantamount to the search for the distribution of firms such that the
implied shape of function dt (s) satisfies the free entry/exit conditions, given the shapeof function ct (s).
Equilibrium Conditions
In order to be explicit about the dependence of dt (s) upon the distribution of firms,
let njt denote the number of firms adopting technology sj . Then, infinite dimensional
15vector nt ≡ {njt }∞j=−∞ represents the distribution of firms in the technology space.

Let cjt ≡ ct (sj ) denote the unit cost of producing a good with technology sj so that
vector ct ≡ {cjt }∞j=−∞ represents the schedule of unit costs. Then, using (17) in (6)15Throughout this paper, we use variables and functions in bold face to represent (infinite dimen-

sional) vectors.

Wants and Past Knowledge January 9, 2006

17

Figure 4: Free entry/exit conditionsand (9), the price indices are written in terms of these vectors:
qt (r) = qb(r; nt , ct ) ≡b t , ct ) ≡Qt = Q(n

"

∞
X

j=−∞

·Z ∞

−∞

¡
¢−(σ−1)
njt cjt eτ |r−sj |qb(r; nt , ct )−(β−1)

dr#−1/(σ−1)¸−1/(β−1)

,

(20)

.

(21)

From (20) and (21), the unit demand price for each sj in (13) can be expressed interms of nt , ct , and population Lt :·b t , ct )dt (sj ) = dbj (nt , ct , Lt ) ≡ Lt Q(n

β−1

Z ∞

−∞

qb(r; nt , ct )σ−β −(σ−1)τ |r−sj |

e

dr

¸1/σ

. (22)

b t , ct , Lt ) ≡ {dbj (nt , ct , Lt )}∞Let d(nj=−∞ be the vector of unit demand prices. Then,the free entry/exit conditions can be stated in the form of a complementary slackness
condition:16
b t , ct , Lt )) · nt = 0, ct − d(nb t , ct , Lt ) ≥ 0, nt ≥ 0.
(ct − d(n(23)

Since ct and Lt are predetermined variables, (23) is a condition that determines
nt in equilibrium at each instant. Figure 4 depicts an example of the distribution offirms that satisfies condition (23). In this example, every firm concentrates at either ofthree points in the technology space, and the unit demand price curve implied by this16

A dot between vectors indicates their inner product; e.g., nt · ct ≡P∞

j=−∞ njt cjt .

Wants and Past Knowledge January 9, 200618

distribution of firms touches the unit cost curve exactly at these three points. Each ofthese points in the technology space is considered as an industry where a continuumof firms monopolistically compete with each other selling goods that are diﬀerentiatedyet aimed at the same consumer want. Except for these three technologies, the unit
demand price is lower than the unit cost and therefore no firm operates.
Using vectors nt and ct introduced above, the equilibrium condition for the labor
RI
market can be briefly stated. The aggregate demand of labor is 0 t `t (i)di while theaggregate supply is Lt . From (14), (18) and (23), each firm produces a unit quantity,st (i)) of labor. Thus, the labor market clears if:
using amount ct (ent · ct = Lt .

(24)

The equilibrium of the economy at each instant is characterized by the distributionof firm nt that satisfies the equilibrium condition for the goods market (23) and one
for the labor market (24). Natural questions are whether such a distribution actuallyexists and, if so, whether it is uniquely determined. In addition, it is of interest to
know the range of technologies that are actually chosen by firms, and the desirabilityof such a choice of technologies in terms of the welfare of consumers. The remainderof this section examines these issues.
Boundedness
This subsection shows that the distribution of firms in equilibrium has a bounded
support, in the sense that no firm chooses technologies extremely distant from others.
Note that function dt (s) in (13) is well defined for all real number s. Diﬀerentiating
it with respect to s and comparing the result with dt (s) gives an upper bound for thecurvature of dt (s):¯¯¯ 1 ∂dt (s) ¯ σ − 1¯¯¯ dt (s) ∂s ¯ < σ τ.

(25)This property can easily be interpreted. Recall that consumers are willing to pay morefor ‘better suited’ goods because such goods provide them with more utility than otherWants and Past Knowledge January 9, 2006

19

poorly matched goods, where the diﬀerence depends on the distance in the technology
space. The unit demand price curve thus cannot have slopes steeper than the valuethat corresponds to this diﬀerence.
Boundedness of equilibrium would be guaranteed whenever the cost of adopting
very distant technology rises with the distance more rapidly than the price that consumers are willing to pay. For the latter claim to be valid, since the production cost is
inversely related to the amount of usable knowledge, it must be the case that firms do
not have good knowledge about very distant technologies. Stated formally, we assume
that
Assumption 1 At date t, there exist a finite value sbt such that kt (s) = 0 whenever|s| > sbt .

Assumption 1 states that the technological frontier is finite in that no firm has adopted
technologies outside [−bst , sbt ] by date t. Under assumption 1, the diﬀerence in costs

provides a centripetal force strong enough to make all firms choose from a bounded

set of technologies.
Proposition 1 Suppose that assumption 1 holds. Then, condition (23) implies that
there exists a finite st > sbt such that:njt = 0 whenever |sj | > st .

(26)

Proof: in Appendix.
Observe that, since st > sbt , Proposition 1 permits the possibility that firms choosea technology that has not been adopted by any firms; i.e., the technology frontier
may advance over time. At the same time, however, it says that the new frontier isbounded since st is finite. The latter property is necessary for proving the existence

and uniqueness of equilibrium.

Wants and Past Knowledge January 9, 2006

20

Desirability
Observe from (8), (20) and (21) that the instantaneous utility of consumers is thereciprocal of the average price index Qt and that the index is determined by nt andct . Thus, given ct , the consumer benefits from a pattern of the distribution of firmsthat minimizes the average price index. Here we show that the distribution of firms
in instantaneous equilibrium is desirable in the sense that it actually minimizes theaverage price index under a certain resource constraint.
b t , ct ) with respect to nt under
Let us consider the problem of minimizing Qt = Q(nresource constraint (24). In addition, we impose condition (26) to this problem in orderto reduce the number of technologies that must be considered in the problem. This
minimization problem can be solved by the standard Kuhn—Tucker method, yielding
the following proposition.
Proposition 2 Suppose Assumption 1 holds and consider the problem of minimizing
b t , ct ) under (24) and (26) with respect to nt ≥ 0. Then, the necessary and suﬃcientQ(ncondition for the solution is (23) and (24).
Proof: in Appendix.

Since condition (26) is implied by condition (23) as shown by Proposition 1, (26)
is no longer required for defining the minimizing, or the derisible, allocation. Recall
that the distribution of firms, nt , constitutes an equilibrium if and only if it satisfies
conditions (23) and (24). Therefore, Proposition 2 states that the equilibrium distribution of firms exactly coincides with the desirable distribution that minimizes theaverage price index (and hence maximizes instantaneous utility) under conditions (24)and (26).
Existence and UniquenessNow we are ready to derive the main result of this section, which relies on the following
proposition.

Wants and Past Knowledge January 9, 2006

21

Proposition 3 The solution to the minimizing problem in Proposition 2 uniquely exists.
Proof: in Appendix.
Given the coincidence between the solution and the equilibrium, Proposition 3 implies
that the instantaneous equilibrium is also unique and existent. Note that the minimizing problem depends (only) on ct and Lt . It means that the solution to the problem,
which coincides with the instantaneous equilibrium, is a function of ct and Lt . Therefore, the equilibrium distribution at each instant can be written as nt = n∗ (ct , Lt ).

5

Dynamics

The analysis in the previous section showed that, given the size of population, theequilibrium distribution of firms at each instant is uniquely determined by the patternof unit costs. Section 3 described how the distribution of firms in turn aﬀects thecosts of production in the future through accumulation of knowledge. This section
examines the dynamic interaction between the distribution of firms and the pattern ofaccumulated knowledge.
Write kjt ≡ kt (sj ) and let kt ≡ {kjt }∞j=−∞ represent the distribution of knowledge
in the technology space. Then (16) implies that the pattern of unit costs is a function
c(kt ). Using these notations and equations (15),of kt and can be written as17 ct = b(18) and (23), the dynamics of the economy can be described as an autonomous systemin terms of kt and Lt :k̇t = n∗ (bc(kt ), Lt ) − δkt ,17

L̇t = λLt .

(27)

−1
The jth element of function bc(kt ) is defined by bcj (kt ) ≡ 1 + ((Tkt )j ) , where T is a matrix

representing knowledge spillover whose (j, m) element is given by Tjm = e−ν|sj −sm | . Note that wetreat vector kt (and any other vector) as a column vector and (Tkt )j denotes the jth element ofvector Tkt .

Wants and Past Knowledge January 9, 2006

22

Given an initial stock of knowledge k0 and the initial size of population L0 , equationsin (27) determine the evolution of knowledge, and therefore the equilibrium path ofthe distribution of firms, for all t ≥ 0.
This section analyzes the dynamic evolution of the economy in three steps: first, wederive the equilibrium distribution of firms at time 0 given an initial stock of knowledge
on one technology. Second, we examine when and how this initial pattern of firm
distribution changes. Third, the evolution of the economy thereafter is investigatedwith the aid of numerical simulations.
Equilibrium at Time 0
Suppose that the initial population L0 is endowed with a small amount of ‘innate’knowledge about using one technology.18 Let this technology be s0 = 0 without loss ofgenerality. Then, each element of the initial stock of knowledge k0 is given by:
k00 > 0, and kj0 = 0 for all j 6= 0,(28)

Substituting (28) for (16) yields the initial unit costs c0 , whose elements are:c0 (sj ) = c0j = 1 + k00 −1 eν|sj | .

(29)

Observe that the unit cost is lowest at technology s0 and therefore, as far as production
cost is concerned, every firm has an incentive to choose it.
To find the equilibrium distribution of firms at t = 0, suppose that every firmchooses technology s0 . Then, the labor market clearing condition (24) requires that n0
should be:
n00 = L0 /(1 + k00 −1 ), and nj0 = 0 for all j 6= 0.
18

(30)

It is obvious from (16) that if k0 = 0, no production activity takes place and therefore experience

never accumulates in our model. Thus, the economy must start with a positive kj0 for at least one
technology. How kj0 emerges in the first place is an interesting problem, but it is beyond the scopeof this paper.

Wants and Past Knowledge January 9, 2006

23

Substituting (29) and (30) into (20), (21) and (22) gives an explicit expression for theb 0 , c0 , L0 ):unit demand price d(n¢¡d0 (sj ) = dbj (n0 , c0 , L0 ) = 1 + k00 −1 ψ(sj ), where
"R ∞
#1/σ
exp {(1 − σ)τ |r − s| + (σ − β)τ |r|} dr−∞R∞ψ(s) =.
exp {(1 − β)τ |r|} dr
−∞

(31)

Function ψ(s) is well defined for all real numbers s, and is a smooth and symmetricfunction with the following properties.
Lemma 1 (i) ψ(0) = 1. (ii) 1 < ψ(s) < exp{((σ − β)/σ)τ |s|} < eν|s| for all s 6= 0.
Proof: in Appendix.
Property (i) implies that, for any value of k0 , the unit demand price at s0 = 0 is
equalized to the unit cost, confirming that each firm choosing technology s0 exactly
breaks even. The remaining equilibrium condition is that the unit demand price mustnever exceed the unit cost for any other technologies. From property (ii) of Lemma 1,(29) and (31), this condition is expressed as:
k00 ≤eν|sj | − ψ(sj )≡ κ(sj ) for all j 6= 0.
ψ(sj ) − 1(32)

The following proposition guarantees that condition (32) is satisfied if the initiallyendowed experience is suﬃciently small.
Proposition 4 (i) lim|s|→0 κ(s) = lim|s|→∞ κ(s) = ∞.
(ii) There exists a finite κ > 0 and s∗ ∈ s \ s0 such that κ = κ(s∗ ) = minj6=0 κ(sj ).
Proof: in Appendix.
Property (ii) of Proposition 4 means that, if and only if k00 ≤ κ, the one-industry
structure represented by (30) satisfies the free entry/exit conditions (23) and therefore
is the unique instantaneous equilibrium. We assume that k00 ≤ κ.
Emergence of New Industries
Given the initial distribution of knowledge (28) with k00 ≤ κ, the economy starts froma one-industry structure, where all firms agglomerate at one point (s0 = 0) in the

Wants and Past Knowledge January 9, 2006

24

technology space. This structure is self-sustaining for a certain time period because
experience accumulates only in that industry, which induces firms in subsequent periods
to agglomerate again at the same technology.
As long as production experience accumulates only on technology s0 and the accumulated knowledge k0t does not exceed κ, the analysis in the previous subsection applies
and, similar to (30), the equilibrium distribution of firms is given by n0t = Lt /(1+k0t −1 )and njt = 0 for all j 6= 0. Substituting this into (27) gives the evolution of knowledge
within the initial industry:−1) − δk0t .
k̇0t = Lt /(1 + k0t(33)

Since population Lt grows at a positive constant rate λ, (33) implies that the amountof knowledge k0t exceeds the critical value κ = κ(s∗ ) in a finite time period. If all
firms continue to choose the initial technology s0 = 0 when k0t > κ(s∗ ), it follows that∗−1
−1 ν|s |)ψ(s∗ ) > 1 + k0te= ct (s∗ ) from (29), (31) and (32). This meansdt (s∗ ) = (1 + k0t

that firms can make positive profits by choosing technology s∗ rather than staying
in the initial industry.19 Therefore, the one-industry structure no longer satisfies thefree entry condition, and firms begin to adopt the new profitable technology s∗ (and−s∗ ). This lowers the unit demand price at s∗ , and the number of firms adopting the
new technology is determined so that the unit demand price at s∗ is equalized to theunit cost there. Note that property (i) of Proposition 4 implies that the newly chosen
technologies (which have the lowest κ(sj )) are significantly diﬀerent, but not extremely
distant, from the currently used technology.
In this way, accumulation of experience in the initial industry eventually gives rise to
new industries that are based on significantly but not extremely diﬀerent technologies.
This structural change can be explained in terms of two forces that aﬀect the technology
choice. In a one-industry economy, the relative unit demand price between a pair ofgoods with diﬀerent technologies is ψ(sj )/ψ(sj 0 ) from (31). Since it is independent of
19

Since function ψ(·) is symmetric around zero, choosing −s∗ is also profitable.

Wants and Past Knowledge January 9, 2006

25

k0t , the centrifugal force induced by the desire of consumers to satisfy a wider rangeof wants is unaﬀected by the accumulation of experience. As experience accumulates,
however, the centripetal force induced by the diﬀerences in the unit cost graduallygets weaker for the following reasons. Recall that equation (16) shows that experience
reduces the unit cost but at a diminishing rate. As the unit cost at the initial industry
approaches the lower bound, which exists at 1, any additional experience within theindustry has little impact on the unit cost there. Spillovers of knowledge, however,significantly reduce the costs of somewhat distant technologies where there still remains
a relatively large gap between the current unit cost and the lower bound. As a result,
the relative cost between the initial industry and other technologies gradually shrinks,
which weakens the centripetal force operating in the technology space.
The preceding analysis has shown that the centripetal force globally dominates thecentrifugal force while the amount of accumulated experience in the initial industry
is below a certain critical value κ. This global dominance comes to an end when k00
exceeds κ. Nonetheless, the centripetal force is locally dominant in the neighborhoodof the initial industry because the unit cost curve (29) has a downward kink at s = 0
while the unit demand price curve (31) is smooth.20 On the other extreme, Proposition
1 means that the centripetal force always dominates the centrifugal force at infinitely
distant technologies. Therefore, the new industry emerges at an intermediate distancefrom the initial industry when k0t reaches the threshold.
Evolution of Industrial Structure
After the one-industry structure becomes unstable, we must deal directly with system(27) to track the evolution of the distribution of firms. Although the previous section
c(kt ), Lt ) is well defined, it does not have an explicit representation.
confirmed that n∗ (bThus, it seems sensible at this point to turn to a numerical simulation.
Figure 5 depicts a simulated evolution of the distribution of firms. Black loci in
20

Recall that (25) shows that the curvature of the unit demand price curve is bounded.

Wants and Past Knowledge January 9, 2006

26

Figure 5: Evolution of industries. Since both ct and nt are symmetric around s0 = 0, results
are shown only for s ≥ 0. Black loci represent the technologies with njt > 0, whereas the height of
the gray area above each locus shows the magnitude of njt cjt . Parameters are set to β = 2, σ = 4,
τ = 1, δ = 0.24, ν = 1, λ = 0.024, and ε = 0.01. Initial conditions are k00 = 0.1 and N0 = 0.1.

the figure represent the evolution of the support of firm distribution, i.e., the set oftechnologies that are adopted by a positive measure of firms. Observe that the economy
consists of a finite number of distinguishable industries, rather than one big industry
that contains a wide range of technologies. Being induced by the centripetal force of
past knowledge, all firms in each industry agglomerate in one point in the technology
space, which creates a strong hysteresis in the choice of technology. However, as
the diminishing returns in learning set in, the centripetal force gets weaker and is
eventually dominated by the centrifugal force generated by consumers’ wants when
new industries emerge. Such a process is repeated cyclically, and as a result structuralchanges occur sporadically at distinct points in time. Thus, interacting forces of wants
and past knowledge create a discrete system of industries while variations in theirrelative strength cause discrete structural changes, even though the underlying modelWants and Past Knowledge January 9, 200627

does not assume such discreteness.21
In Figure 5, the height of the gray area above each locus represents the size of eachindustry measured by consumer expenditure, or equivalently the number of workers inthat industry. Observe that after emergence the size of an industry initially rises andthen falls,22 for the following reasons. At the birth of an industry, the cost of adoption
is very high, and the demand for the highly priced goods is low even though they wellmeet wants that are unsatisfied by existing industries. As experience accumulates,
the price falls and the demand increases. Since there remains a wide margin for cost
reduction in new industries the prices of goods fall more rapidly than those in theolder industries through the accumulation of experiences, which eventually makes the
new industry larger than typical existing industries. However, the demand for the newindustry begins to decrease when a yet newer industry emerges, which better serves
some wants that were previously served by the original industry. In the long run, someindustries approach their steady state sizes, while other industries disappear.
Disappearance of some industries might require a further explanation. To see this,note that when a frontier industry emerges, it attracts a large unsatisfied demand fromthe wants beyond the frontier. Thus, it may be profitable to adopt a technology 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 technology too close to other industries on both sides. In some21

Strictly speaking, the model for convenience assumes that firms choose technologies from an infi-

nite but discrete set s ≡ {sj }∞j=−∞ , where sj = jε, with ε > 0 being a very small constant. However,
we confirmed that, given that ε is suﬃciently small, the size of ε does not aﬀect the simulation results,
which implies that the discrete structural changes in the model do not depend on the discreteness ofs.
22This pattern is consistent with the finding by Wang (2004, Figures 3,4), who examined theevolution of the relative industry GDP (i.e., each industry’s share of the GDP) for seven industries in
the U.S. and found that the relative industry GDP initially rose and then fell, with only one exception:
the nylon industry.

Wants and Past Knowledge January 9, 2006

28

cases, the demand eventually becomes so small that no firm in this industry can breakeven. Are, then, such ephemeral industries futile from the viewpoint of economic
growth? In fact, they play an important historical role: the knowledge accumulated
by such industries spills beyond them and serves as a stepping stone to new, possibly
everlasting, frontier industries.

6

Growth Cycles

In the previous section, simulation results illustrate that industries sporadically emerge
and disappear as the economy grows. We now investigate how such continual changes
in industrial structure aﬀect the pattern of macroeconomic growth rate and the utilityof the representative consumer.
Fluctuations in the GDP Growth Rate
In equilibrium, the total output quantity of goods with technology sj is njt whereas
their price is cjt . Therefore, the per capita real GDP growth rate (i.e., the real GDP
growth minus the population growth) in a conventional definition is given by:23
gt =ṅt · ct− λ.
nt · ct(34)

Diﬀerentiating both sides in (24) with respect to time gives ṅt · ct + nt · ċt = L̇t .
Applying it to (34) enables us to express the growth rate in terms of the rate of cost
reduction:

23

¶µ
¶
∞ µ
X
nt · ċt
njt cjtċjt=−.
gt = −
nt · ct j=−∞
Lt
cjt

(35)

Equation (34) implicitly assumes that vector nt changes continuously. This can be confirmedin the following way. As shown by Proposition 2, nt coincides with the solution to the problem
b t , ct ) under (24) and (26) with respect to nt . The constraint of this problem isof minimizing Q(ncompact and changes continuously as ct moves according to (36). Then we can apply the Theoremof the Maximum (Stokey and Lucas 1988, Chapter 3) to be assured of the continuity of nt .

Wants and Past Knowledge January 9, 2006

29

Figure 6: The per capita real GDP growth and its decomposition. The overall per
capita real GDP growth rate is represented by the thick curve, whereas the thin curves give its
decomposition into contributions from each industry according to formula (35). Parameter values
and initial conditions are the same as in Figure 5.

Note that njt cjt /Lt represents the market share of industry j. Thus, the last expressionin (35) gives a clear decomposition of the growth rate in each industry: gt is theweighted sum of the rates of cost reduction in each industry, where the weights are
their market shares.
Equation (35) enables us to calculate numerically the evolution in the GDP growthrate, the result of which is shown in Figure 6. We can observe three conspicuous features of the figure. First, at each point in time usually only one or two, at most three,young industries explain quite a large portion of the overall growth rate, as Gordon(2000) pointed out in the case of computer and related industries. In other words, mature industries contribute little to economic growth, implying that structural changes,
especially the emergence of new industries, are crucial for sustained economic growth.
Second, the figure shows that the rate of economic growth falls immediately after a new
industry emerges. That is, structural changes cause a temporary slowdown in growthalthough they are indispensable in the long run. This phenomenon corresponds to theWants and Past Knowledge January 9, 200630

well-known “productivity slowdown puzzles” (see Hornstein and Krusell 1996; Greenwood and Yorukoglu 1997; Jovanovic and Rousseau 2005). Third, the variation in
the rate of economic growth is larger in the early stage of development and graduallystabilizes as the number of industries increases. Such a tendency was mentioned by
Lucas (1988), and subsequently confirmed by data (e.g., Blanchard and Simon 2001;
Koren and Tenreyro 2005).
In the model, the last property is easy to interpret: as the number of industries
increases, the contribution of the rise and fall of each industry to the economy-widefluctuations in growth becomes smaller, and eventually the law of large numbers will
apply. To understand the first two properties, we must explicitly look at the patternof knowledge accumulation that drives economic growth. Recall that the rate of cost
reductions in each industry is influenced by the pattern of knowledge accumulation,
which in turn depends on the distribution of firms. Diﬀerentiating (16) with respect
to time and then applying (27) to it yields:ċt = (Ct − I)2 Tnt − δ(ct − 1),(36)

where T is a symmetric matrix with its (j, m) element being Tjm = e−ν|sj −sm | , Ct ≡diag(ct ) is a diagonal matrix whose jth diagonal element is cjt , I is an identity matrix,
and 1 is a vector with all elements equal to 1. Substituting (24) and (36) into (35)gives the per capita real GDP growth in an intuitive form:
2−1
gt = L−1
t nt · (Ct − I) Tnt − δ(1 − Lt 1 · nt ).

(37)

The second term in (36) shows the extent to which depreciation of knowledge decelerates growth. As long as the rate of depreciation δ is not large, the significance of thisterm is limited.
The main body of gt is determined by the first term, which represents the eﬀect ofknowledge accumulation through production experience on the macroeconomic growthrate. This term can be viewed as a quadratic form in nt with (Ct − I)2 T being the matrix of coeﬃcients. Since (Ct − I)2 is a diagonal matrix, any interaction eﬀect between

Wants and Past Knowledge January 9, 200631

industries is described by matrix T. Note that, reflecting the fact that experience in acertain industry is more useful within that industry than to other industries, the diagonal elements in T are larger than its oﬀ-diagonal elements. This suggests that there
is an agglomeration economy: with a large market share an industry can accomplish afast cost reduction, which in turn accounts for a large portion of the macroeconomic
growth rate. When a new industry emerges, the individual market shares of industries
become more diverse. The emergence of the new industry thus diminishes the benefitof agglomeration and lowers the rate of economic growth, at least in the short run
during which matrix (Ct − I)2 can be regarded as approximately constant.24
In the long run, however, variations in matrix (Ct −I)2 play a crucial role in equation(37). This matrix implies that mature industries where cjt − 1 is near zero contributelittle to economic growth. Sustained growth thus requires continual emergence of new
industries, and the new industries must adopt unexplored technologies in which there
still remains room for further improvement. The learning-by-doing process within anindustry aﬀects the growth rate only transitorily, while spillover of knowledge acrossdiﬀerent technologies contributes to economic growth in the long run by paving theway for the emergence of ever newer industries.25
Welfare Eﬀects
We have shown that, in terms of GDP, emergence of new industries temporarily decelerates growth, although they are necessary for long-term growth. GDP is a convenientmeasure, but it does not account for the benefit resulting from increases in the vari-

24

Note that vector nt can change instantly, while ct cannot.

25

In this respect, our model contrasts clearly with and is complementary to the ‘hybrid’ endogenous

growth models presented by Young (1998), Peretto (1998), and Dinopoulos and Thompson (1998),
which combine the variety expansion model and the quality ladder model. In their models, the primary
determinant of long-term growth is quality improvement within each industry, while the expansion ofvariety is treated as a transitory adjustment process that does not aﬀect the long-term growth rate.

Wants and Past Knowledge January 9, 200632

ety of goods. We now examine the evolution of the consumer’s instantaneous utility
and show that it also follows the same cyclical pattern as the GDP growth during theprocess of economic growth.
Note that instantaneous utility ut is the reciprocal of the average price index Qt as
shown by (8). Thus, using (21), the rate of change in ut can be decomposed into:bc (nt , ct )bn (nt , ct )
u̇t
Q̇t
Q
Q
=−
=
· (−ċt ) −· ṅt ,
b t , ct )b t , ct )
ut
Qt
Q(n
Q(n

(38)

bn (·) denote the partial derivatives of function Q(·).
bb c (·) and Q
The first term
where Qin (38) represents the welfare improvement that comes from the falling prices of goods.

After some calculation and applying the equilibrium condition (23), we can show that
this term coincides with the per capita GDP growth gt . The second term in (38)represents the benefit from the increased variety of goods. Calculation shows that it
equals (gt + λ)/(σ − 1); that is, (σ − 1)−1 times the rate of GDP growth–not percapita GDP growth.26 Intuitively, the increased aggregate purchasing power enables
more firms to enter the market, which has a positive pecuniary externality on every
consumer who values more variety.
Collecting these eﬀects together, we now have a simple formula:
σ
λ
u̇t
=
gt +
.
ut
σ−1
σ−1

(39)

Equation (39) implies that the utility growth of households evolves in parallel with per
capita real GDP growth. Thus, the argument we have expounded with respect to the
GDP growth applies also to the utility of households; the successive emergence of new
industries is indispensable to maintain a sustained utility growth (above λ/(σ − 1)),
but the rate of utility growth is temporarily reduced immediately after each industry
emerges.
26

Since the size of firm is constant in equilibrium, the increased purchasing power proportionally

increases variety. Substituting (20) and (21) into (8), we see that the electricity of instantaneous
utility with respect to variety (i.e., nt ) is (σ − 1)−1 . Then the result follows.

Wants and Past Knowledge January 9, 2006

7

33

Conclusion

Consumers desire to satisfy an indefinite range of wants. This creates a large demand
for goods made with novel technologies that meet the unsatisfied needs of consumers.
However, firms face a high cost in adopting such previously unexplored technologies
because relatively little is known about them. We investigate how this tradeoﬀ between
demand and cost, or between wants and knowledge, aﬀects the distribution of firms in
the technology space. We find that firms endogenously form a number of distinguishable industries and their dynamic evolution is characterized by the sporadic emergenceof new industries. The emergence of a new industry temporarily reduces the rate of
per capita GDP growth, because it diversifies the share of individual industries anddiminishes the benefits of agglomeration economy that comes from knowledge accumulation within an industry. New industries are, however, indispensable for sustained
growth since without them cost reduction within existing industries eventually comesto an end. We also confirm that the instantaneous utility of the consumer, which incorporates the benefit from increased variety, also follows the same pattern as the percapita GDP growth.
In the literature of endogenous growth, the majority of studies stressed the supply
side (e.g., accumulation of knowledge) as the determinant of growth. However, ourresult highlights the importance of consumer demand in directing the economy to
the right way to sustained economic growth.27 Economic growth can be maintained
only when knowledge is accumulated over an ever-expanding range of technologies. We
show that firms willingly pay high costs of adopting previously unexplored technologies,
even though they cannot appropriate knowledge generated by their behavior. Rather,
their incentive is induced by the demand of consumers who desire to satisfy an ever-27

The importance of consumer demand in directing growth is also examined by Foellmi and

Zweimüller (2004). They show that inequality may promote growth since rich consumers are able
to satisfy a wider range of wants, although growth cycles are not their focus.

Wants and Past Knowledge January 9, 2006

34

expanding range of wants.28 The interaction between knowledge accumulation on the
supply side and human wants on the demand side is thus an indispensable part of thesustained growth.

Appendix
Proof of Proposition 1
Consider the case of s > sbt . From Assumption 1 and (16),X0e−ν|s−s | kt (s0 ) = e−ν(s−bst )

s0 ∈s

X
s0 ∈s

0

e−ν(bst −s ) kt (s0 ) = e−ν(s−bst ) (ct (bst ) − 1)−1 .

Substituting it for (16) gives:st ) − 1)eν(s−bst )ct (s) = 1 + (ct (bfor s > sbt .

(A.1)

for s > sbt .

(A.2)

Note that dt (bst ) ≤ ct (bst ) from condition (23). Then, (25) implies that:dt (s) ≤ ct (bst )e((σ−1)/σ)τ (s−bst )st ) > 1 is predetermined and since ν > ((σ − 1)/σ)τ , there must be a finite
Since ct (bvalue st > sbt such that the right-hand side of (A.1) is strictly larger than that of (A.2)
for all s > st . This means cjt = ct (sj ) > dt (sj ) = djt for all sj > st , from which
st .
condition (23) implies njt = 0 for all sj > s. A similar argument applies for s < −b
Proof of Propositions 2 and 3t. Note that, since
Since st is finite, we can choose a Jt such that [−st , st ]∩ s = {sj }Jj=−Jtb t , ct )−(β−1) .
b t , ct ) is equivalent to maximization of Q(n1 < β < σ, minimization of Q(nFrom (20) and (21), this equivalent problem can be written as:#(β−1)/(σ−1)
Z ∞"X
Jt
¡
¢
τ |r−sj | −(σ−1)
max
njt cjt edr,n−Jt t ,...,nJt t

28

−∞

(A.3)

j=−Jt

This property is consistent with a finding by Acemoglu and Linn (2004) that the size of potential

demand is critical for the entry of new drugs and pharmaceutical innovation.

Wants and Past Knowledge January 9, 200635

subject to (24). Observe that condition (26) is implicitly incorporated into problem(A.3). We set up a Lagrangian for this problem and diﬀerentiate it to obtain thefirst order condition. Let the Lagrange multiplier on (24) be ξt . Then, the resulting
complementary slackness condition is:Z ∞−1 β − 1ξtqb(r; nt , ct )σ−β e−(σ−1)τ |r−sj | dr ≤ cσjt
σ − 1 −∞

(A.4)

for all |j| ≤ Jt and with equality if nj > 0.

Substituting (A.4) into (24) and solving it for ξt gives:Z
1 β−1 ∞
ξt =qb(r; nt , ct )−(β−1) dr.
Lt σ − 1 −∞(A.5)

Substituting (A.5) back into (A.4) shows that the first order condition for problem(A.3) is exactly the same as (23).
The domain for {n−Jt t , . . . , nJt t } defined by (24) and (26) is compact and the object
function in (A.3) is continuous and strictly concave in those variables. The existence
and uniqueness of the solution follow from these properties (thus Proposition 3 holds).
In addition, those properties guarantee that the first order condition (23), combinedwith constraints (24) and (26), is both necessary and suﬃcient. In fact, constraint (26)
is not necessary to define the minimizing solution since it is implied by (23) and (24)from Proposition 1. This completes the proof of Proposition 2.
Proof of Lemma 1
For convenience, define function ζ(r, s) = (σ − β)τ |r| − (σ − 1)τ |r − s|, by which ψ(·)
in (31) can be expressed as:"R ∞exp ζ(r, s) dr−∞ψ(s) = R ∞−∞exp ζ(r, 0) dr

#1/σ

.

(A.6)

Property (i): With expression (A.6), it is obvious that ψ(0) = 1 holds since thenumerator and the denominator coincide when s = 0.
Property (ii): Consider the case of s > 0. It is straightforward to confirm ζ(r, s) ≤ζ(r − s, 0) + (σ − β)τ s with strict inequality when r < s. Substituting this inequality

Wants and Past Knowledge January 9, 200636

into (A.6) shows ψ(s) < exp {((σ − β)/σ)τ s} . From 1 < β < σ and ν > ((σ − 1)/σ)τ ,it follows that ν > ((σ − β)/σ)τ . Thus, we obtain ψ(s) < eνs . Similarly, it can be
confirmed that ζ(r, s) ≥ ζ (r − ((σ − 1)/(β − 1))s, 0) with strict inequality if 0 < r <((σ − 1)/(β − 1))s. Substituting this into (A.6), we have ψ(s) > 1. Collecting bothresults shows 1 < ψ(s) < exp {((σ − β)/σ)τ s} < eνs for all s > 0. Since ψ(s) issymmetric around zero, s can be replaced by |s|.
Proof of Proposition 4Note that function κ(s) defined in (32) is well defined for all real number s except at
s = 0, and is symmetric, smooth, and positive for all s 6= 0. Since it is symmetric, it
is suﬃcient to consider only the case of s > 0.
Property (i): When s tends toward zero from above, both the denominator andnumerator of κ(s) also tend to zero. Applying l’Hopital’s Theorem, we have:ν − ψ 0 (s)
ν−0
=
= ∞,
s→+0
ψ 0 (s)0lim κ(s) = lim

s→+0

where we used ψ 0 (0) = 0 since ψ(s) is everywhere diﬀerentiable and symmetric aroundzero. When s tends to infinity, we use Lemma 1 and divide both the denominator andnumerator of κ(s) by exp{((σ − β)/σ)τ s} to obtain:
exp{(ν − ((σ − β)/σ)τ )s} − 1
∞−1
=
= ∞.
s→∞
1 − exp{−((σ − β)/σ)τ s}
1−0

lim κ(s) ≥ lim

s→∞

(A.7)

Property (ii): Note that, since sj = jε and ε > 0, (A.7) also imply limj→∞ κ(sj ) =∞. In addition, κ(sj ) > 0 for all j > 0. Thus, there exists a smallest elementκ = κ(s∗ ) > 0 in sequence {κ(sj )}∞j=1 .

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