Discussion Paper No. 560

ENDOGENOUS GROWTH AND CYCLESWITH A CONTINUUM OF TECHNOLOGIES

Ryo Horii

October 2001

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

Endogenous Growth and Cycles
with a Continuum of Technologies1
Ryo Horii2
October 24, 2001

1

I would like to thank Kenn Ariga, Masahisa Fujita, Shinsuke Ikeda, Kazuo Mino, Yoshiyasu

Ono, Akihisa Shibata and seminar participants at Tokyo University, Kyoto University and the2001 spring meetings of the Japanese Economic Association for valuable comments and discussions. All errors are naturally my own.
2

Correspondence: Ryo Horii, Institute of Social and Economic Research, Osaka University,

6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. <horii@iser.osaka-u.ac.jp>

Abstract
When a continuum of technologies is introduced to the model of Grossman and Helpman(1991), both continuous and discrete technological progress may occur as a result oftechnology choices by private firms. A good is created through R&D based on one of acontinuum of technologies that diﬀer in productivity, and the R&D cost is smaller when
there is greater public knowledge about that technology, which accumulates throughspillovers. When firms shift continuously to superior technologies, there is no incentive
to retain existing technologies and the economy grows smoothly. By contrast, when many
firms choose the same technology, accumulated knowledge makes this choice privately
optimal for a certain time period, and the economy grows cyclically through a sequence of
discrete progresses in technology. These two dynamics constitute multiple equilibria, andit depends on the size of the parameters which equilibrium is desirable for consumers.
JEL Classification Numbers: E32, O33, O41.
Keywords: endogenous growth; growth cycles; technology choice; R&D spillovers;
continuous and discrete technological progress; general purpose technologies.

1

Introduction

Technology choice involves a trade-oﬀ between the initial cost of investment and the
productivity and/or the quality of the product. It is one of the most important problems
for firms and entrepreneurs. In addition, the pattern of technology choices by individual
firms is a matter of great significance at the level of the macroeconomy, since adoptionof superior technologies contributes not only to the total factor productivity of the economy, but also to the stock of public knowledge through various types of spillovers. Undercertain conditions, the augmented public knowledge promotes the adoption of even more
superior technologies at the level of individual firms, which in turn accumulates publicknowledge of these new technologies. This micro-macro interaction brings about endogenous technological progress, which is one of the most important sources of sustained
economic growth.
In the last decade, a considerable number of endogenous growth models have beenpresented that have investigated technological progress due to spillovers of knowledge.
Most of these studies have focused on the dynamics where technology advances continuously.1 However, as economic historians have pointed out, there has been not only
continuous technological progress but also discrete advances in technology, such as theintroduction of steam engines, electricity, and computers.2 In these cases the pattern ofgrowth was not necessarily smooth. Partly in response to these historical facts, some
economic theorists have developed models which are capable of explaining this kind of
discrete technological change and the resulting cyclical patterns in economic growth.3
While most existing papers focus exclusively on either continuous or discrete technological progress, this paper shows that both patterns of technological progress may berealized as multiple equilibria when we explicitly model the technology choices by private
firms from a continuum of technologies. We will clarify the way in which new technologies are chosen and when they are adopted, and derive a path of economic growth that is
substantially diﬀerent depending on the patterns of technological progress. In addition,
our analysis makes it possible to compare the desirability of continuous and discrete

1

Seminal works include Romer (1990) and Grossman and Helpman (1991).

2

See Helpman (1998) for references.

3

Representative works include Bental and Peled (1996), Helpman and Trajtenberg (1998) and Free-man, Hong and Peled (1999). See also Horii (2000) for cyclical growth arising from discrete changes in
the industrial structure.

−1−

technological progress in terms of welfare.
This paper develops an endogenous growth model that generalizes the basic varietyexpansion model of Grossman and Helpman (1991, hereafter GH) by allowing firms tochoose from technologies that diﬀer in productivity. Specifically, we extend the GH
model in the following three respects.
1. There exists a continuum of technologies, and each firm can conduct R&D based on
one technology to create a good. The marginal cost of production diﬀers dependingon the technology choice.
2. The labor cost of R&D on a certain technology is reduced by the accumulation of
public knowledge on that technology. However, this eﬀect gets weaker when thecost becomes suﬃciently small.
3. R&D conducted by each firm contributes not only to the stock of public knowledge
on the technology adopted by this firm, but also to the stock of knowledge on
related technologies, according to the degree of association.
In this setting, firms confront a trade-oﬀ in choosing technologies. On the one hand,
they have an incentive to choose technologies that have higher productivity, since this will
enlarge the firm’s profit stream. On the other hand, it is advantageous to choose technologies that are similar to those adopted by existing firms, since such technologies can be
adopted with a small initial cost by virtue of the large stock of public knowledge. These
two forces represent centrifugal force and agglomeration force in the technology space,respectively. Their relative magnitude determines the equilibrium pattern of technology
choice in R&Ds. Subsequently, the trade-oﬀ itself is dynamically aﬀected, through theprocess of knowledge accumulation. This creates an endogenous evolution in the patternof technology choice and hence economic growth.
We will demonstrate that after a short time period of take-oﬀ, the economy entersa period of sustained growth. In this period, there are two patterns of equilibrium
dynamics, which we call the ‘continuous growth regime’ and the ‘cyclical growth regime,’
respectively. In the continuous growth regime, technology choices by new firms at eachdate shift continuously in the direction of higher productivity. That is, the technology
adopted by new firms today is slightly more advanced than that adopted yesterday, whichis slightly more advanced than that adopted the day before yesterday, and so forth. Dueto the R&D spillovers across technologies, the costs of R&D using superior technologies−2−are gradually falling, and thus it is privately optimal to adopt a technology slightly moreadvanced than that adopted yesterday, given that all the other firms follow this patternof technology adoption. The speed of technological progress cannot be infinite since themagnitude of R&D spillovers between quite diﬀerent technologies is small. In fact, theequilibrium speed of technological progress is determined so that the marginal benefit ofchoosing a technology with higher productivity exactly equates to the marginal cost of
adopting a technology with less public knowledge.
The dynamics in this regime diﬀer from the original GH model in at least three respects. First, growth is sustained, not by exponential increases in the variety of products,
but by steady improvements in the production technology. The economy will converge
to a balanced growth path where both the cost of R&D and the number of new goods
introduced at each instant become constant. Second, in this paper the rate of economic
growth depends heavily on the magnitude of knowledge spillover across technologies.
This factor is not considered in the GH model but is believed to have some significancein the real economy. Third, two economies that initially diﬀer in their performances may
or may not converge to the same state, depending on their initial structures. It is evenpossible that their relative performance will eventually be reversed.
While the above dynamics fully satisfy the equilibrium conditions, another quite
diﬀerent pattern of dynamics exists which still satisfies all of the conditions for a perfectforesight equilibrium. Suppose that the same technology is adopted by all entrepreneurs
during a certain time period.
The cost of R&D declines throughout this period through spillovers of knowledge, but
the rate of cost reduction is not uniform across technologies. In general, a non-negligible
portion of the knowledge gained from experiences with a certain technology is specificto that technology, and cannot be applied to other technologies. This implies that thecost of R&D based on this technology is reduced more rapidly than R&D based on othertechnologies. Thus, provided that other entrepreneurs adopt the same technology, it isprivately optimal to choose this technology.4 Moreover, an additional adoption of thistechnology makes the cost of R&D even lower, resulting in a chain of adoption of thesame technology by new entrepreneurs. As knowledge accumulates and the technology
becomes mature, however, the R&D cost becomes less sensitive to marginal increases
in knowledge, while it becomes possible to adopt increasingly advanced technologies at
a given labor cost owing to spillovers across technologies. In fact, when the number
4This economy of scale is the primary cause of multiple equilibria.

−3−

of firms adopting the same technology reaches a certain threshold, it becomes more
profitable for entrepreneurs to switch to a significantly advanced technology. With the
adoption of the new technology, the rate of economic growth temporarily declines. It
then gradually accelerates as knowledge on the new technology is accumulated throughthe R&D experiences, creating a cyclical pattern in the rate of economic growth.
This pattern of technology adoption and its resulting fluctuations in output is essentially similar to those in the models of general purpose technologies (GPTs),5 among
which Helpman and Trajtenberg (1998, hereafter HT) is the closest. Between our modeland the HT model, however, there is a crucial diﬀerence in the process of the emergenceof GPTs. The HT model assumes that GPTs are developed somewhere outside themodel, and once a new GPT is introduced exogenously firms immediately switch to the
new technology. By contrast, our model shows that GPTs are endogenously formed as aresult of technology choices by private firms. Consequently, which technologies become
GPTs depends on the equilibrium path. In addition, our model shows that even thougha technology is available that has higher productivity than the current GPT, firms do
not switch to it until the cost of adoption becomes suﬃciently small through spilloversof knowledge between the current and superior technologies.
In terms of welfare, the formation of a GPT is beneficial in that it reduces the costof R&D and promotes the introduction of a large variety of products, especially by theaccumulation of technology-specific components of public knowledge, which are neverutilized in the continuous growth regime. However, the formation of a GPT also has
drawbacks. Once a GPT is formed, it creates a discrete diﬀerence between the R&D
cost based on it and those based on other technologies. This makes entrepreneurs stickwith the current GPT until a technology with suﬃciently high productivity becomes
available at a certain cost of adoption. Moreover, when there is a significant diﬀerencein productivity between the current GPT and the potential new technology, only a small
portion of the experience with the current GPT can be applied to the new technology.
This implies that the average speed of technological progress tends to be slower in thecyclical growth regime compared to the continuous growth regime. We will show thatit depends on the size of the parameters whether or not the benefits flowing from the5

According to Lipsey, Bekar and Carlaw (1998), a technology is qualified as a GPT if it has scopefor improvement, a wide variety of uses, a wide range of use, and strong technological complementarities
with existing or potential new technologies. In fact, it will be shown that each technology adopted inthis regime satisfies all these conditions.

−4−formation of GPTs exceed the disadvantages.
The rest of the paper is organized as follows. Section 2 presents the model and derives
the instantaneous equilibrium. Section 3 investigates the evolution of R&D activities andshows that the dynamics of economic growth can be divided into a transitory take-oﬀ
period and a stage of long-term growth. Having analyzed the dynamics in the take-oﬀ
period, sections 4 and 5, respectively, derive two regimes of dynamics in the long-term
growth period, which are described above intensively. Section 6 investigates which regime
is desirable by comparing welfare in the two regimes. Section 7 concludes the paper. The
proofs of all claims are collated in the appendix.

2

The Model

2.1

Firms and Technologies

In the model, there are many firms competing monopolistically, and each firm manufactures a single variety of diﬀerentiated goods.6 The number of firms is determined inequilibrium through the process of free entry. To set up a new firm, an entrepreneur
needs to invest in R&D to develop a new diﬀerentiated product and its manufacturing
process. At the time of entry, each firm chooses its own production technology from acontinuum of available constant-returns-to-scale technologies.7 Depending on technology
choice, the quantity and/or quality of goods that can be produced from a given amountof labor diﬀer. For ease of notation, we normalize the quantity of each good so that every
good enters the utility function symmetrically, and thus all diﬀerences are expressed byproductivity. Each technology is indexed by a positive number z ∈ (0, ∞) such thatwhen technology z is adopted this firm can produce amount z (1−α)/α of its product froma unit of labor, where α ∈ (0, 1) is a constant which will be defined in equation (4).8

Although in principle any technology z ∈ (0, ∞) can be adopted, the labor cost of

6

Like most variety expansion models, we assume that the potential variety of products is suﬃciently

large that it does not become exhausted on the path of economic growth. We also assume that imitating
an existing product costs no less than creating a new variety of diﬀerentiated good, for both technicaland legal reasons. Thus, entrepreneurs have no incentive to imitate existing products and thus each goodis produced by exactly one firm.
7

We assume that once a technology is adopted firms will not change their production technology since

doing so is as costly as creating another new diﬀerentiated good.
8

This method of indexing simplifies the mathematical expressions to come, but it is not essential.

−5−R&D diﬀers depending on the technology choice. The labor input required to develop
a new product is small when a large stock of public knowledge already exists on thetechnology adopted. Specifically, we assume that to create a blueprint of a new good
based on technology z, an entrepreneur must devote c(Kt (z)) units of labor to R&D.9
Here Kt (z) is the stock of public knowledge with respect to technology z, and c(·) is acost function defined in (0, ∞) which satisfies the following properties.
Assumption 1 c(·) satisfies
a. c(k) > 0 and c0 (k) < 0 for all k ∈ (0, ∞),b. The elasticity of c(k), ²c (k) ≡ (k/c(k))c0 (k), is strictly decreasing in k,c. limk→+0 ²c (k) ≥ 1 and limk→∞ ²c (k) = 0.10
While our specification of the R&D cost is based on Romer (1990) and GH,11 we incorporate two additional realistic features. First, rather than focusing on aggregate “generalknowledge,” we explicitly deal with the heterogeneous distribution of public knowledge
across various technologies. Second, while GH assumed that the R&D cost is reduced
at a constant rate when knowledge (or experience) accumulates, we incorporate the tendency that the R&D cost becomes less sensitive to marginal increases in knowledge as
the technology becomes mature.
The public knowledge accumulates as a by-product of R&D activities. When a firm
creates a blueprint based on technology z0 , it augments the stock of knowledge Kt (z0 )by one unit due to the positive technological spillover ‘within’ the technology z0 . Thiscondition is the same in the GH model. In our model, there are, in addition, positive
spillovers ‘across’ technologies depending on the degree of association between them: forany technology z > 0, knowledge Kt (z) is augmented by s(z0 , z), which is assumed tosatisfy the following properties.12

9

A more formal statement is that an entrepreneur who devotes l units of labor to R&D for a timeinterval of length dt can create blueprints of lc(Kt (z))−1 dt new goods that are based on technology z.
10

In fact, it is suﬃcient to assume a much weaker condition, limk→+0 ²c (k) > 1/δ and limk→∞ ²c (k) <1/δ, where δ > 1 is a constant that will be introduced in claim 1
11Note that when c(k) = a/k and Kt (z) = Kt for all z, our specification coincides with that of GH.

12

For simplicity, we assume that the magnitude of spillover between the two technologies depends only

on their productivities.

−6−Assumption 2 For any z0 > 0 and z > 0, s(·, ·) satisfies
a. s(z0 , z) ∈ (0, 1) and s(z0 , z0 ) = 1,
b. s(z0 , z) = s(z, z0 ),c. s(z0 , z) = s(αz0 , αz) for all α > 0,d. s(z0 , z 0 )s(z 0 , z) = s(z0 , z 00 )s(z 00 , z) for all z 0 , z 00 ∈ (z0 , z),e. limz0 →∞ (z 0 − z0 )s(z0 , z 0 ) = 0.
The first three items, a, b and c, specify respectively boundedness, symmetry and homogeneity of the spillover function s(·, ·). Property d assumes that the spillovers are
consistent in that, for any given z0 and z, ‘the portion of experience in technology z0

that can be applied to some intermediate technology’ multiplied by ‘the portion of experience in that technology that can be applied to z’ is not changed by the choice of theintermediate technology. The final property e, means that spillover eﬀects are not large
between significantly diﬀerent technologies.
Under these restrictions, the functional form of s(·, ·) can be expressed in terms oftwo parameters.

Claim 1 There exist µ ∈ (0, 1] and δ > 1 such that s(z0 , z) = µ min{z/z0 , z0 /z}δ for all

z0 and z 6= z0 .

In this expression, parameter δ measures the rate at which the magnitude of spillover getssmaller as the diﬀerence between the productivities of two technologies increases, andµ indicates the portion of the component of public knowledge that is not technologyspecific. From claim 1 and assumption s(z, z) = 1, the whole process of knowledge
accumulation is summarized byK̇t (z) = µ

Z ∞
0

½ 0
¾
z z δ
min
dRt (z 0 ) + (1 − µ)(Rt (z) − Rt (z−)),
,
z z0

(1)

where Rt (·) is the cumulative distribution function of R&D at time t. That is, Rt (z)represents the instantaneous flow of new products introduced at time t that are based ontechnologies whose productivity is lower than or equal to z, and Rt (z)−Rt (z−) represents
the flow of new products based exactly on technology z.13 We assume that the public
13

Rt (z−) represents the limiting value of Rt (z 0 ) when z 0 approaches z from below.

−7−knowledge does not depreciate, in which case there is a one-to-one relationship between
the distribution of knowledge and the distribution of firms.
½ 0
¾
Z ∞
z z δ
min
dFt (z 0 ) + (1 − µ)(Ft (z) − Ft (z−)),,
Kt (z) = µ
z z0
0

(2)

where Ft (z) represents the cumulative distribution function of firms in the technology
space z ∈ (0, ∞).

2.2

Consumers and the Market Demand

The demand side of our model is essentially the same as the GH model. There are L
identical consumers. Each consumer inelastically supplies one unit of labor service atthe prevailing wage wt at every date and maximizes utility over an infinite horizon,Z ∞Ut =e−ρ(τ −t) ln Dτ dτ.
(3)tHere ρ is the subjective discount rate, and Dt represents an index of consumption attime t. Consumers have tastes for variety, expressed by Dixit and Stiglitz (1977) typepreferences for bundles of diﬀerentiated goods,
∙Z Jt
¸1/α
α
Xt (j) dj
,
Dt =

(4)

0

where parameter α ∈ (0, 1) represents substitutability between goods, Jt ≡ Ft (∞) de-notes the number of goods produced at time t, and Xt (j) the consumption of good j.

For the same reason as in the GH model, growth may cease in economies that are
too small (small L), too impatient (large ρ) and/or too elastic in terms of substitution
between goods (large α), since R&Ds are not privately profitable in such economies.
Because of space constraints, we focus exclusively on economies where growth can besustained, though this is not essential to the results of the analysis.
Assumption 3 (1 − α)L/αρ is greater than Γ, where Γ is a finite constant depending

only on functional forms of c(·) and s(·, ·). The definition of Γ is presented in the
appendix.

Now, we turn to the market demand for goods. Let Pt (j) denote the price of good j.
Preference (4) implies that given the spending Et at time t the maximized value of Dt
is Et /P̄t , where P̄t is a price index of diﬀerentiated goods,P̄t =∙Z J tα0 − 1−αP (j )

0

−8−

dj

0

¸− 1−αα.

Here, the consumer’s problem reduces to maximizingZ ∞e−ρ(τ −t) (ln Eτ − ln Pτ ) dτ.
Ut =t

Since the above expression is separable in Et and Pt , it is obvious that the optimal
expending rule satisfies Ėt /Et = rt − ρ. We follow the GH model by normalizing prices
so that aggregate consumption expenditure, LEt , becomes unity at every instant. In thiscase, the nominal interest rate rt equals the subjective discount rate ρ for all t, and themarket demand for good j can be calculated as
α1LXt (j) = Pt (j)− 1−α P̄t1−α .

2.3

(5)

Instantaneous Equilibrium

In the settings presented above, we can characterize the instantaneous equilibrium of
this economy at a certain time t, given the current and future distribution of firms, Ft0 (·)
for all t0 ≥ t. At this point, we impose a simple assumption on the distribution of firms
for all t.14

Assumption 4 For any t, supp Ft (·) is bounded andR∞0

zdFt (z) is finite.

Consider a firm j which has already developed a blueprint based on a certain technology z. Since this firm can produce amount z (1−α)/α of its good from a unit of labor,
the marginal cost of production is wt /z (1−α)/α . Given the demand function (5), the firm
maximizes operating profits
∙Z Jt
¸−1
´
³
1
α
− 1−α
(1−α)/α0 − 1−α0Pt (j )
dj
Pt (j)Pt (j) − wt /z
0

by charging a price Pt (j) = wt /(αz(1−α)/α).

When all firms follow the above rule, each firm’s quantity and operating profits are
determined by its technology z, the current wage wt and the current distribution of firmFt (·).
αz 1/αR∞,
wt 0 z 0 dFt (z 0 )(1 − α)zπt (z) = R ∞ 0.
0
0 z dFt (z )xt (z) =

14

(6)
(7)

Here, supp Ft (·) means the support of Ft (·), that is, the set of technologies that have been adopted

by some firms by t. In fact, it is suﬃcient to assume that the initial distribution satisfies the property
given in assumption 4, since the dynamic analysis to follow will show that when the economy starts withsuch a distribution it will never become inconsistent with assumption 4.

−9−

These equations characterize the equilibrium of the goods market.
Labor services supplied by consumers are applied to the production of diﬀerentiatedgoods and R&D. From (6), the labor demand for production workers is calculated as
α/wt . The labor market equilibrium requiresα+ LR
t = L,
wt(8)

where the second term represents the total employment in R&D, defined as
Z ∞
R
Lt ≡
c(Kt (z))dRt (z).

(9)

0For simplicity, we rule out speculative bubbles so that the stock market value of a
firm equals the present discount value of its profit stream,15
Z ∞vt (z) =e−ρ(τ −t) πτ (z) dτ.

(10)

t

Under the assumption of free entry, the value of a firm or equivalently, the value of its
blueprint, must not be higher than the cost of creating a new blueprint with the same
technology.16 On the other hand, firms invest in R&D based on technology z only whenthe value of the blueprint based on that technology is not less than the cost of creating
it. The following free entry condition summarizes these requirements:
vt (z) ≤ wt c(Kt (z))
for all z ∈ (0, ∞) and with equality on the support of Rt (·).

(11)
Note that Kt (·) is apredetermined function, and vt (·) is also given since we take the future distribution offirms as given.
To find the equilibrium wage level satisfying this condition, we introduce an index of
the labor cost of R&D when a certain technology is chosen. Let Zt denote the frontier
technology, that is, the maximal element in the support of Ft (·), and let Vt denote thevalue of a current frontier firm, vt (Zt ). Then, the value of a new blueprint based on
an arbitrary technology z can be expressed by (z/Zt )Vt , since equation (7) and (10)imply that the profit stream of a firm and thus its discounted sum are both linear in its
technology level z. Using this fact, we can construct the following index,
lt (z) ≡Zt
c(Kt (z)),
z

(12)

15

Here we utilize the fact that rt = ρ holds for all t.

16

Otherwise, an infinite number of firms would want to enter into R&D, leading to excess demand in

the labor market, which in turn would lift the wage rate until the above condition is met.

− 10 −

which represents the amounts of R&D labor required to create a bundle of new blueprints
based on technology z that has in total the same value as a current frontier firm. Thereis a bounded set of technologies with which the required labor input takes the minimum
value.
Claim 2 ltmin ≡ minz∈(0,∞) lt (z) exists, and the set of minima, Lmin
≡ {z| lt (z) = ltmin },t

is bounded.

Using this index, condition (11) reduces to a simple form,
Vt ≤ wt lt (z).

(13)

The above expression shows that the demand curve for R&D labor is horizontal at
the level of Vt /ltmin . Recall also that the demand curve for production labor, α/wt , isdownward sloping and crosses the inelastic supply curve at wt = α/L. Since the totallabor demand is the sum of these two, the equilibrium wage is determined as
wt = max{Vt /ltmin , α/L}.

(14)

Note that the demand for R&D labor is positive if and only if the equilibrium wage islarger than α/L, and that only the technologies that give the smallest lt (z) are adopted
for R&D,
supp Rt (·) ⊂ Lmin
t .

(15)

The arguments so far have demonstrated that once the current and future expectations of the distribution of firms is given, instantaneous equilibrium conditions specify
prices pt (·), quantities xt (·), profits πt (·), and the prevailing wage wt uniquely, and also
specify the support of the R&D distribution Rt (·). For this instantaneous equilibrium toconstitute a part of a dynamic equilibrium path, however, evolution of the distributionof firms must be consistent with actual R&Ds invested in at each instant,Ḟt (z) = Rt (z).

(16)

That is, a path of Ft (·) constitutes a perfect foresight equilibrium if and only if the R&D
distribution Rt (·) implied by (16) satisfies the labor market clearing condition (8) and thefree entry condition (15) at all t. Once an equilibrium path of Ft (·) is found, economic
growth is measured by the rate of growth in consumption,¶(1−α)/αµZ Zt
αzdFt (z)
,
Dt =
wt L
0
− 11 −(17)

which is obtained by substituting (6) for (4). In the following sections, we investigate
evolutions in the amount of R&D investments and the technology choice in the R&D,
which determine the path of Ft (·) and hence economic growth.

3

Evolution of R&D Activities

At each date, entrepreneurs face two problems: they must decide whether or not to invest
in R&D, and, if they invest, they must also choose the technology. Though this decision
cannot be completed until the future expectation on the R&D activities is specified, thissection shows that some essential properties of the evolution of R&D activities can beinferred from just the current and the past state of the economy.
First, we focus on the decision on whether or not to invest. Entrepreneurs invest
in R&Ds only when the discounted sum of the future profits, Vt , is no smaller than
the investment cost, wt ltmin . Note that the profits become small when the number ofcompetitors rises. This is expressed by πt (Zt ) = (1 − α)/Nt , where Nt is a weighted sum
RZof the number of firms, Nt ≡ Zt−1 0 t z dFt (z).17

Since no firm exits the goods market, the amount of profits a firm can get never rises

after entry. Thus, Vt must be no larger than (1 − α)/ρNt , and this maximum value isrealized only if there is no subsequent entrance of competitors. On the other hand, thecost of R&D is no smaller than αltmin /L since there is a lower bound on the wage rate
at wt = α/L. Note also that this minimum R&D cost is realized only when there isno entry at time t. From these facts we can derive the exact condition under which a
positive number of entrepreneurs invest in R&D.
Claim 3 At each date in a perfect foresight equilibrium,a. If there is a positive amount of R&D, then Vt ∈ (αltmin /L, (1 − α)/ρNt ). Otherwise,
Vt = (1 − α)/ρNt .

b. There is a positive amount of R&D if and only if (1−α)/ρNt is larger than αltmin /L.
Next, we turn to the issue of technology choice. When an entrepreneur chooses froma continuum of technologies, he or she will take into account two facts. On the one
17

Note that when all firms use the same technology, such as in the GH economy, index Nt coincides

with Jt . However, when some firms use a technology whose productivity is lower than the current frontier
technology, the number of these firms is discounted according to the value of z relative to Zt .

− 12 −

hand, as shown in (7), the profit stream to a firm is magnified proportionally to z whenit adopts a technology with higher productivity. This provides an incentive to choose atechnology significantly superior to past ones. On the other hand, it is advantageous tochoose a technology with a large stock of public knowledge since such a technology can be
adopted with a small initial cost. This creates an incentive to choose a technology that issimilar to those chosen by past entrepreneurs, a dynamic agglomeration force. Note thatas the stocks of knowledge on existing technologies become large, the agglomeration force
gets weaker, since the R&D cost becomes less sensitive to a marginal diﬀerence in theamount of knowledge between technologies. The equilibrium pattern of technology choice
is determined by the relative magnitudes of the centrifugal force and the agglomeration
force.
Whenever a new technology is adopted, the centrifugal force and the agglomeration
force must exactly balance at that technology level, otherwise adopting either more advanced or more conservative (less advanced) technologies will provide higher profitability.
In fact, there is a certain level of knowledge under which these two forces cancel out.
Claim 4 If there is a technology z ∈ supp Rt (·) such that Zt0 < z holds for all t0 < t,then Kt (z) = K̄ ≡ ²−1c (1/δ).

Note that the above claim says only that the amount of knowledge must be K̄ when
the technology frontier advances. For example, suppose that the number of firms in theeconomy is quite small and that no technology has amount K̄ of knowledge. Then, theimplication of claim 4 is not that there is no R&D, but that the frontier does not advance
unless knowledge accumulates up to K̄.
Here, another question is whether the frontier advances whenever the amount ofknowledge reaches K̄?18 The answer depends on both parameters and the current distribution of firms. Suppose that µ is less than unity and there is a positive mass offirms which have adopted the same technology. Then, equation (2) implies that there isa portion of accumulated knowledge that is specific to this technology. In this case, entrepreneurs may want to adopt this established technology even when knowledge exceeds
K̄, since adoption of even a slightly advanced technology will cause discrete increases
in the R&D cost. While this creates a ‘lock in eﬀect’, the following claim shows thatthe adoption of some advanced technologies eventually becomes more profitable when asuﬃciently large amount of knowledge is accumulated. It also shows that the amount of
18

This is equivalent to the question whether the reverse of claim 4 holds or not.

− 13 −

knowledge at the frontier never exceeds K̄ when the frontier technology is not a masspoint 19 or µ = 1.
Claim 5a. Let c̃(k) ≡ c(K̄)(K̄/µk)1/δ . Then, there exists a finite value K max such that c̃(k) <c(k) for k > K max and c̃(k) > c(k) for k ∈ [K̄/µ, K max ).

b. min supp Rt (·) > z holds whenever Kt (z) ≥ K max .
c. The minimum element of supp Rt (·) that is not a mass point in Ft (·) is greaterthan z whenever Kt (z) > K̄.
d. If µ is unity, min supp Rt (·) > z holds whenever Kt (z) > K̄.
Though the exact dynamics cannot be specified by claim 4 and 5, they suggest that
it is useful to break down the process of economic growth into two periods, a transitory
take-oﬀ period and a period of sustained growth. Let us define the take-oﬀ period as
the time period before the technology frontier, Zt , starts to advance. During this period,
economic growth is primarily led by reductions in R&D costs caused by accumulation ofknowledge on existing technologies, since only a limited set of technologies are adopted.20
This period lasts until the amount of knowledge on the initial frontier technology reaches
some threshold value, which exists between K̄ and K max and depends on the initial distribution of firms. After the take-oﬀ, entrepreneurs start to adopt superior technologies.
Throughout this period of sustained growth, the amount of knowledge on any technology on which new products are created is bounded between K̄ and K max . Since thisboundary is finite, there is no long-run trend in the amount of knowledge and hence inthe cost of R&D. Thus, in the period of sustained growth, only technological progress
explains long-term growth.

3.1

Dynamics in the Take-oﬀ Period

The rest of this section investigates the dynamics of the economy in the take-oﬀ period.
We first derive the motion of macroeconomic variables in this period, assuming that the19

In what follows, we call a technology which is adopted by a positive mass of firms, mass point in

distribution Ft (·). In mathematical terms, z is a mass point in Ft (·) if and only if Ft (z) − Ft (z−) > 0

holds.
20

Note that, however, this process cannot sustain economic growth permanently because, as the numberof firms grows, the returns to R&D will decline more rapidly than the cost of R&D declines.

− 14 −

economy starts from a quite simple distribution of firms.21 Subsequently, the condition
for the end of this period is derived.
Suppose that initially only one technology is adopted by all existing firms to produce
their goods. Let us denote the number of initial firms by n0 and, without losing generality,
let the initial technology be unity.
F0 (z) = 0 for z < 1,

n for z ≥ 1.
0

(18)

Given the initial distribution, subsequent evolution of Ft (·) is determined by R&D activities of entrepreneurs. In the following, we characterize these dynamics by focusingR
on two macroeconomic variables, Nt ≡ Zt−1 zdFt (z) and Vt ≡ vt (Zt ).
From equation (2), the amount of knowledge available on the initial technology, Kt (1),coincides with the number of firms, Nt , throughout the take-oﬀ period. Since all R&D
investments are based on technology z = 1 within this period, ltmin = lt (1) = c(Nt )holds whenever LR
t > 0. Applying this equation to (14), the equilibrium wage becomes
wt = max {Vt /c(Nt ), α/L} . The flow of new goods developed at each instant is Ṅt , andeach R&D activity requires c(Nt ) units of labor. Thus, the total amount of R&D laborRis LR
t = c(Nt )Ṅt . Substituting these expressions for wt and Lt into the labor market

equilibrium condition (8) yields a law of motion for Nt ,½¾LαṄt = max
− ,0 .
c(Nt ) Vt(19)

On the other hand, derivative of equation (10) provides a standard arbitrage condition
for the value of each firm, v̇t (z) = ρvt (z) − πt (z) for all z. When z is set to unity, thiscondition and equation (7) specify the dynamics of Vt in terms of Nt and Vt itself,
1−αV̇t = ρVt −
.
(20)
Nt
The above two equations characterize the dynamics of the economy in the take-oﬀ
period, and enable us to draw a phase diagram in (Nt , Vt ) space. Setting Ṅt = 0 andV̇t = 0 in equations (19) and (20), respectively, we have two curves on which either Nt
or Vt stays constant,
NN :
VV :
21

αc(Nt )
,
L
1−α
Vt =
.
ρNt
Vt =

(21)
(22)

As will become clear in the following sections, the long-term dynamics are not aﬀected by the

specification of the initial distribution.

− 15 −

Figure 1: Phase diagram in the take-oﬀ period.
The take-oﬀ period ends when entrepreneurs adopt some technology z whose productivity is higher than that of the initial technology. Recall that claim 4 says Kt (z) must
be exactly K̄ when the frontier advances. For this condition to be satisfied, Nt must belarger than K̄/µ and the new technology must be z = (µNt /K̄)1/δ . Thus, the exact con¡¢dition of take-oﬀ can be found by comparing lt (µNt /K̄)1/δ = c(K̄)(K̄/µNt )1/δ = c̃(Nt )with lt (1) = c(Nt ). In fact, claim 5 shows that c̃(Nt ) becomes smaller than c(Nt ) whenNt exceeds K max .
Figure 1 shows the locations of N N curve and V V curve, along withSS :αc(K̄)
Vt =
L

µ

K̄
µNt

¶1/δ

for Nt > K̄/µ,(23)which represents the minimum cost of R&D by using technology z = (µNt /K̄)1/δ . In
the region of 0 < Nt < K max , all entrepreneurs adopt the initial technology. The statevariables follow (19) and (20), which we call the take-oﬀ period dynamics, until Nt , orKt (1), reaches the critical value K max . When Nt reaches K max , choosing either z = 1or z = (µNt /K̄)1/δ yields the same rate of return, and then the economy immediately
enters the period of sustained growth.22 The dynamics in the period of sustained growth,which will be analyzed in the following sections, determine the value Vt at the end of
22

Note that Kt (1) is strictly increasing in time as long as growth does not cease, which is guaranteed− 16 −

the take-oﬀ period.23 Subsequently, the transitional dynamics in the take-oﬀ period are
specified by solving (19) and (20) backwards from this point.
If N0 is larger than K max , the return from investing in technology (µNt /K̄)1/δ isbetter than the returns from choosing the initial technology. Note also that, from claim
3, R&Ds are carried out if and only if V V curve locates above SS, which is equivalenttoN0 < N+≡µ(1 − α)L
αρc(K̄)¶δ/(δ−1) ³µ ´1/(δ−1)
.
K̄

(24)

Thus, if the number of initial firms is in an intermediate range of (K max , N + ) the economy
enters the period of sustained growth immediately, with V0 settling somewhere between
SS and V V . In an economy with too many existing competitors, however, profits are sosmall that no entrepreneur wants to invest in any R&D and thus no growth occurs. In
the rest of this paper we assume (24).

4

Continuous Growth Regime

After the take-oﬀ period, the pattern of technology adoption and thus that of economic
growth may not be unique. However, there is a special case in which the dynamics canbe fully specified. Suppose that parameter µ is unity. Then, claim 4 and part d of claim
5 show that the amount of its knowledge Kt (Zt ) is always kept at K̄ and only the frontier
technology is adopted by new entrepreneurs at each instant. This property implies thatwhenever there is a positive amount of R&D, the technology frontier, Zt , must advance
continuously without creating any mass point in Ft (·).24
In an economy where µ is smaller than unity, it may be possible that a mass of firms
adopt the same technology, enabling them to utilize technology-specific components ofknowledge. However, for the moment we suppose that no mass point is created in acertain time period. In such a situation, the second term in equation (1) vanishes and thepattern of knowledge accumulation becomes substantially equivalent to the case of µ = 1.
This implies that it is optimal for private entrepreneurs, who take knowledge distribution
under assumption 3. Then, the part b of claim 5 implies that no entrepreneur adopts technology z = 1
after Kt (1) reaches K max .
23

It follows from claim 3 that this value must settle somewhere between N N and V V . Under assump-tion 3, N N necessarily locates above V V when Nt = K max .
24

Note that if a positive mass of firms adopted the same technology, then Kt (Zt ) exceeds K̄.

− 17 −

as given, to act similarly as in the economy with µ = 1. That is, the entrepreneurs choose
continuously advanced technologies without creating any mass point, consistent with theinitial assumption. This consideration shows that, regardless of the value of µ, there isan equilibrium path which follows the continuous pattern of technology adoption. Below
we derive the equilibrium dynamics in this ‘continuous growth regime.’
Let us assume that any mass point does not emerge during the time interval in which
the frontier locates in a certain interval [Z S , Z E ).25 Combining claim 4 with part cof claim 5, we can confirm that the frontier continuously advances and only frontier
technology is adopted in this regime, which corresponds to the µ = 1 case. When there
is no mass point, we can express Ft (·) in terms of a density function f (·),
S

Ft (z) = Ft (Z ) +Z min{z,Zt }f (z 0 ) dz 0 .

ZS

for z ∈ [Z S , Z E ).

(25)

Note that the value of f (z) at each z does not depend on t, since it is only when Zt = z
that new firms enter the market with technology z and thereafter the density at z does
not change. For the amount of knowledge at the frontier to coincide with K̄ at eachdate, density function f (·) must satisfy the following equation.
µ

Zt
ZS

¶−δ

K̄ + µZ Zt µ
ZS

Zt
z0

¶−δ

f (z 0 ) dz 0 = K̄

for Zt ∈ [Z S , Z E ).

(26)

Diﬀerentiating both sides with respect to Zt and conducting some transformation, equation (26) turns out to be equivalent tof (z) =

δ K̄µz

for z ∈ [Z S , Z E ).

(27)

Once the density of firms becomes clear, we can derive the speed at which the newtechnologies are adopted, using the labor market equilibrium conditions. Note thatltmin = c(K̄) holds since only the frontier technology is adopted. Substituting it for (14),©
ªthe equilibrium wage turns out to be wt = max Vt /c(K̄), α/L . The flow of the totallabor input into R&D is calculated by substituting the above wage rate into the labormarket clearing condition (8). On the other hand, from equation (27) we can calculate
the cumulative input of R&D labor that is required for the technology frontier to move¡¢
RZ
from ZS to Zt , as Z St c(K̄)f (z 0 ) dz 0 = (δ K̄c(K̄)/µ) ln Zt − ln Z S . Diﬀerentiating thisexpression with respect to t and equating it to the derived flow of labor input, we have
25

Here, Z E may be infinite.

− 18 −

the growth rate of the technology frontier.
µ½¶ ¾αc(K̄)
Żt
µ
gt ≡
L−
= max
,0 .
Zt
Vt
δ K̄c(K̄)(28)

This shows that the technology frontier advances if and only if Vt is greater than a certain
value V̂ ≡ αc(K̄)/L.

With equation (28) in hand, we can express the dynamics as an autonomous system

of Nt and Vt in a similar fashion to our expression for the take-oﬀ period dynamics.
When Ft (·) is expressed in the form of (25), where f (·) is given by (27), Nt becomes´³R SZ
Zt −1 0 z dFt (z) + (δ K̄/µ)(Zt − Z S ) . Diﬀerentiating this equation with respect to t,
yields a simple expression Ṅt = (N ∗ − N )gt , where N ∗ ≡ δ K̄/µ is a stationary point of

Nt . This equation can be expressed in terms of Nt and Vt by applying equation (28).
µ
¶

αc(
K̄)
µ


L−
if Vt > V̂ ,(N ∗ − Nt )
V
δ
K̄c(
K̄)
t
Ṅt =
(29)


0 if Vt ≤ V̂ .

As we have seen in the above, when Vt is less than or equal to V̂ , no firms enter themarket and thus Nt does not change. When Vt is greater than V , new firms enter themarket and the technology frontier advances. While the entry of new firms increases thetotal number of firms, the advance of the frontier decreases the relative profitability ofexisting firms to new frontier firms, which makes Nt small. Equation (29) shows thatthe overall eﬀect on Nt is positive when Nt is less than N ∗ , while it is negative when
Nt > N ∗ .
On the other hand, a fundamental rule of diﬀerentiation allows the time derivativeof Vt ≡ vt (Zt ) to be deconstructed into two parts,V̇t = v̇t (Zt ) + vt0 (Zt )Żt ,(30)

where the first term represents the eﬀect from variation in the value function vt (·), while
the second term is derived from the advance of the frontier.
The standard arbitrage condition gives the first term, v̇t (Zt ) = ρVt −(1−α)/Nt . Sincevt (z) is equivalent to (z/Zt )Vt , its derivative is simply Vt /Zt . Further, from the definitionof gt , Żt = Zt gt . Substituting these into formula (28), we have V̇t = (ρ + gt ) Vt − (1 −α)/Nt . Then using equation (28), the dynamics can again be expressed in terms of Nt− 19 −

Figure 2: Phase diagram in the continuous growth regime.
and Vt ,
µ


 ρ+

¶
µL
1−α
αµ
Vt −
−
Nt
δ K̄c(K̄)
δ K̄
V̇t =

1−α

 ρVt −
if Vt ≤ V̂ .
Nt

if Vt > V̂ ,(31)

By equating the above expression to zero, we also have a locus on which Vt does notchange.

VV :

µ


 ρ+

¶−1 µ
¶
αµ
µL
1−α
if Vt > V̂ ,+Nt
δ K̄c(K̄)
δ K̄
Vt =


 1 − α if Vt ≤ V̂ .
ρNt

(32)

If actual Vt is below this locus, Vt must fall to keep the arbitrage condition, while Vt rises
when Vt is above the locus.
Equation (29) and (31) characterize the dynamics of (Nt , Vt ) and enable us to draw aphase diagram. Figure 2 depicts the phase diagram, where the dotted curve represents the
maximum possible value of a frontier firm, (1− α)/ρNt , and the horizontal line at Vt = V̂expresses simultaneously the minimum possible cost of R&D, αltmax /L, and the ṅ = 0
locus. The configuration of these two curves shows that there is a positive amount of R&D− 20 −

so long as Nt is smaller than the intersection of these two curves, N̂ ≡ (1−α)L/αρc(K̄).26

With the starting value of Nt and the final value of Vt specified, the dynamics of Nt andVt within this regime is uniquely determined, and they also specify the dynamics of Zt
since (28) provides a one-to-one non-decreasing correspondence between Vt and the speedof technological progress. Specifically, when the economy stays in this regime forever, theinitial value of Vt must be on the stable arm that converges to a saddle point (V ∗ , N ∗ ),
where
V∗ ≡

µc(K̄)
δρK̄c(K̄) + µL

(33)

represents the value of a frontier firm on the balanced growth path.27 On the balanced
growth path, the technology frontier advances at a constant rate ofg∗ =(1 − α)µL− αρ > 0.
δ K̄c(K̄)(34)

Once the movements of Nt , Vt and Zt are derived, it follows from equations (14) and(17) that consumption evolves according to
Dt =

4.1

αc(K̄)(Zt Nt )(1−α)/α .
LVt

(35)

Comparison with the GH model

In this subsection, we compare the dynamics of our model in the continuous growth
regime with that of the original GH model, where only one technology is adopted. This
will make clear the diﬀerence caused by the introduction of multiple technologies, themodified R&D cost function, and spillovers across technologies.
First, let us focus on the workings of each model that make economic growth sustainable. In the GH economy, long-term economic growth is achieved through exponential
increases in the flow of new goods. Such an explosion in variety is due to the assumption
26

As we have seen in the previous section, the first technology adopted in the period of sustained

growth is (µN E /K̄)1/δ , where N E ∈ [K max , N + ) denotes the final value of Nt in the take-oﬀ period.
Then, the starting value of the normalized number of firms is determined as Nt = N E /(µN E /K̄)1/δ =(K̄/µ)1/δ N E(δ−1)/δ

. Applying the fact that K max ≥ K̄/µ and the value of N + given by equation (24),the result is that Nt resides within a finite interval [K̄/µ, N̂ ). In addition, it will become apparent thatonce this period starts with Nt ∈ [K̄/µ, N̂ ), it will not get out of this interval whenever Kt (Zt ) = K̄.
Note also that N ∗ also resides in [K̄/µ, N̂ ) under assumption 3.
27

Otherwise the economy will enter the region below V̂ or the one above the dotted curve, contradicting

claim 3 either way.

− 21 −

that the labor cost of R&D is reduced at a constant rate as knowledge accumulates.
By contrast, the economy in our model cannot grow through this mechanism because
the cost becomes insensitive to a marginal increase in knowledge when it has become
suﬃciently small. Actually, the number of new goods introduced at each instant is constant on the balanced growth path, but the quantity (which can also be interpreted as
quality) of each new product is steadily increasing because continuously more advanced
technologies are adopted to produce the goods. Due to spillover eﬀects, the cost of R&D
based on every technology is declining, but at the same time entrepreneurs are willing to
adopt increasingly advanced technologies which require large R&D labor inputs. In equilibrium, these two eﬀects exactly cancel out and the cost becomes constant. Empiricalobservations show no exponential trend in the number of R&D projects or exponential
decreases in the labor cost of R&D. Consequently, the latter story, expressed by ourmodel, seems more plausible.
The second concern is the rate of economic growth, especially on the balanced growthpath. The GH model specified the cost function by c(k) = a/k,28 and derived the growthrate of consumption index Dt as
∗
=
gD,GH

(1 − α)2 L− (1 − α)ρ.
aα

To make the growth rate in our model comparable with the above equation, suppose thecost function of R&D takes a specific form c(k) = a/k + b, which is one of the simplest
functional forms satisfying assumption 1.29 The rate of economic growth on the balanced
growth path is then calculated by applying (34) for equation (35),
∗
=
gD

1−α ∗
µ (1 − α)2 L
g = 2− (1 − α)ρ.
α
δ
aα

(36)

It is observed that the only diﬀerence between the two expressions is the inclusion of µ/δ 2
in the first term of (36). That is, all the elements that enhance the growth rate in the GH
model will also enhance the growth rate in our model. In addition, our model shows thatthe growth rate is also influenced by δ and µ, both of which concern the magnitude of28

Interestingly, if we weaken assumption 1 and let c(k) = a/k in our model, the dynamics completely

coincide with that of the GH model. Thus assumption 1 is crucial in the realization of technologicalprogress.
29

The small positive constant b can be interpreted as a fixed cost of R&D investment that cannot be

eliminated by the accumulation of public knowledge. Note, however, that the assumption 1 does notnecessarily require the existence of such a fixed cost.

− 22 −

R&D spillover eﬀects across technologies. Specifically, when the larger portion of R&D
experience can be applied to superior technologies, the term µ/δ becomes larger andhence the growth rate increases. This eﬀect seems to have some relevance in the realprocess of economic growth.30
Third, we turn to the issue of convergence. In contrast to the neoclassical growthmodels that predict the convergence of economies, the GH model predicts that the proportional gap between economies will remain unchanged forever. Specifically, in the GH
model, economies instantly jump to the balanced growth path and grow at the same
constant speed. This absence of transitional dynamics is a by-product of the assumptionof a one-to-one correspondence between the number of competitors (which determines
the amount of profits) and the labor cost of R&D. Though such a correspondence isuseful for many endogenous growth models to make growth sustainable, the dynamics
in this section show it is not a necessary condition for the sustained growth: given theconstant cost of R&D, c(K̄), the number of competitors, Nt , can vary between K̄/µ andÑ .
The phase diagram in figure 2 shows that Nt will converge to N ∗ in the long run,which seems somewhat like the dynamics in the neoclassical growth models. This does
not mean, however, that convergence in the levels of GDP will occur, since GDP depends
not only on the number of firms but also on their technologies. For example, suppose that
there are two economies, both of which are in the continuous growth regime and diﬀer
only in the normalized number of firms, Nt . In transition, the amount of investments
should be larger in the economy with small Nt since there are few competitors and thus
expected profits are large.31 Note that the diﬀerence in the amount of investments implies
not only convergence in Nt , but also divergence in Zt since the speed of technologicalprogress, gt , is proportional to investments in this regime. Thus, in the first half of thetransition period, the diﬀerence in GDP between the two economies rapidly shrinks forboth of the above reasons. However, before they settle into the balanced growth path,the initial gap in GDP is eventually reversed and, moreover, it is theoretically possiblethat the absolute value of the proportional gap becomes even larger than its initial value.

30

Note that the growth rate does not depend on parameter b as long as it is positive. This implies that

it is inappropriate to accept the GH model as an approximation when there is a positive lower bound in
the labor cost of R&D, however small it is.
31

This can be confirmed in the phase diagram where the stable arm is downward sloping.

− 23 −

Of course, such a reversal does not happen if the initial diﬀerence is not in Nt but in Zt .32
These examples suggest that the micro structure of the economy should be consideredwhen the issue of convergence is analyzed.
Finally, there is yet another diﬀerence between the GH model and our model in thepossibility of multiple equilibria. When µ is smaller than unity, there are significantly
diﬀerent equilibrium dynamics in which technology advances discretely rather than continuously. These will be investigated in the next section.

5

Cyclical Growth Regime

In the previous section, we concentrated on the situation where no mass point emergedin the distribution of firms, and demonstrated that there are equilibrium dynamics inwhich entrepreneurs choose continuously advanced technologies. When the economy
follows such equilibrium dynamics, the technology-specific portion of knowledge is neveraccumulated. Without the opportunity to utilize technology-specific knowledge, there
is no incentive for entrepreneurs to choose an existing technology at each instant. This
explains exactly why no mass point emerges. In this sense, the behavior of each agent
and macroeconomic movements are mutually consistent along this path. However, thisconsideration also suggests that there may be another equilibrium path in which theseare mutually consistent but in a diﬀerent way: when a positive mass of firms choose
the same technology, this choice may become privately optimal for each firm, since they
can utilize the technology-specific portion of accumulated knowledge to reduce R&D
costs. Throughout this section, we focus on the economy where µ is less than unity andinvestigate these sorts of equilibrium dynamics by assuming that entrepreneurs createa sequence of discrete mass points in Ft (·) rather than choosing continuously advanced
technologies.
Suppose that there is a small mass of firms that use the same technology z0 , which isthe frontier technology at the time. When the number of firms adopting technology z0 isn, this technology has amount (1 − µ)n of its specific knowledge. This portion of knowledge makes technology z0 relatively attractive for subsequent entrepreneurs because of

low initial R&D costs. An additional adoption of this technology makes the cost of R&D
even lower, resulting in a chain of adoption of the technology z0 by new entrepreneurs.
32

When the distribution of the two economies are given (18), where n0 diﬀers between them, reversalof the initial gap may or may not happen depending on the parameter values.

− 24 −

Figure 3: Determination of the size and time of technological improvement
However, this chain does not last forever, as shown in claim 5. Entrepreneurs eventuallyswitch to a certain advanced technology, λz0 , when the number of firms adopting technology z0 reaches a certain threshold, n̄. Below, we show the ways in which the size of
improvement and the threthould are determined.
As the number of firms with technology z0 increases, the amount of knowledge on advanced technologies increases due to the R&D spillovers across technologies. Specifically,
if an entrepreneur devotes c(K̄) units of labor,33 he or she can create a good by adopting¡¢1/δ 34z0 .
Since the amount of profit is proportional
a new technology z = µn/K̄ + 1

to the technology choice, the relative profit that can be gained by adopting the new¡¢1/δ, which implies that the incentive to adopt atechnology, rather than z0 , is µn/K̄ + 1

superior technology increases as n gets larger. On the other hand, the labor cost of R&D
based on technology z0 gradually falls due to the spillover eﬀects within this technology.

Specifically, the relative cost of adopting the new technology is c(K̄)/c(K̄ + n), whichmeans that the incentive to stick with technology z0 is also increasing in n.
Figure 3 depicts the variations of relative profit and relative cost as the number of33

Recall that whenever an entrepreneur adopts a technology superior to the current frontier, the amountof knowledge on this technology must be K̄ and thus the labor cost must be c(K̄).
34This equation follows from Kt (z) = K̄.

− 25 −

firms at the frontier increases. It shows that the relative cost dominates the relative
profit for small values of n.35 For these values of n, it is profitable for entrepreneurs tochoose technology z0 and thus the frontier does not advance. However, as the knowledge
on technology z0 grows, the cost of R&D becomes insensitive to a marginal increasein knowledge, which causes a slowdown in the increase of the relative cost. Eventually
the magnitude of relative cost coincides with that of relative profit when n reaches acertain threshold value n̄.36 At this point, entrepreneurs are indiﬀerent as to the choice
between the current or new technology, but after that point the adoption of the newtechnology becomes more profitable.37 Thus, immediately after the number of firms
reaches the maximum value n̄, the technology frontier jumps to the new technology, where
the relative improvement of technology is the same as the relative profit at this time,λ ≡ (µn̄/K̄ + 1)1/δ = c(K̄)/c(K̄ + n̄). Entrepreneurs then continue to adopt technology

λz0 until an even more advanced technology λ2 z0 is adopted, when the number of firms

with technology λz0 reaches n̄. This pattern of technological adoption creates cyclical
growth dynamics, which we call the ‘cyclical growth regime.’
To investigate the movements of macroeconomic variables, we first derive the dynamics within ‘one cycle’, that is, within the time interval throughout which all entrepreneurs
adopt the same technology. Let N S ∈ [K̄/µ, N̂ ) denote the value of Nt at the beginningof the cycle.38 Initially the amount of knowledge at the frontier is K̄ and it increases

with the number of firms at the frontier, which yields Kt (Zt ) = Nt − N S + K̄.39 The

motion of Nt within a cycle is then obtained in essentially the same way as we havederived (19).
Ṅt = max½¾
αL
− ,0 .
c(Nt − N S + K̄) Vt(37)

Note that the above expression includes c(Nt − N S + K̄) not c(Nt ), which is the only
35

The derivative of the relative profit with respect to n at the point n = 0 is K̄/δ, while that of therelative cost is K̄/δµ. Note that the latter is larger as long as µ < 1.
¡
¢1/δ
36
It is straightforward to show that a unique solution exists to µn/K̄ + 1= c(K̄)/c(K̄ + n) in theregion of n > 0.
37

Figure 3 shows that the adoption of the new technology becomes more profitable after n reachesn̄ assuming that only the old technology is adopted. Actually, entrepreneurs start to adopt the newtechnology after that, which makes the profitability of adopting the new technology even higher relative
to the old technology.
38

See footnote 26.

39

Note that the number of firms at the frontier is Nt − N S .

− 26 −

Figure 4: Dynamics within a cycle
diﬀerence between (19) and (37). Accordingly, the Ṅ = 0 locus becomes
NN :

Vt =

αc(Nt − N S + K̄)
.
L

(38)

From the arbitrage conditions we can also obtain the time derivative of Vt and the V̇ = 0
locus, which are exactly the same as (20) and (22), respectively.
Figure 4 depicts the phase diagram of one cycle dynamics, defined by (37) and (20),along with a typical path. As shown in claim 3, the whole path must locate between
the N N and V V curves. This condition is satisfied if and only if the initial value
S (N S ) and V S (N S ), given inof Vt in this cycle, denoted by V S , locates between Vinf
supthe figure.40 When a cycle starts from a state (N S , V S ) where V S satisfies the lattercondition, Nt reaches the critical point N S + n̄ in a finite time.41 The technology frontier
then jumps to a new technology which is λ times as productive as what it was. WithR
the jump of Zt , Nt ≡ Zt−1 z dFt (z) is divided by λ, while Vt ≡ vt (Zt ) is multiplied byλ. This whole process constitutes a one-to-one mapping from the state of the economy
40

S
S
Here, Vinf
(N S ) and Vsup
(N S ) are obtained by following the dynamics backward from the points(N S + n̄, V̂ /λ) and (N S + n̄, (1 − α)/ρ(N S + n̄)), respectively. From this procedure, it is obvious thatS
S
V̂ < Vinf(N S ) < Vsup
(N S ) < (1 − α)/ρN S holds.
41

This is guaranteed under assumption 3.

− 27 −

Figure 5: Phase diagram of Φ(·, ·)at the beginning of a cycle, (N S , V S ), to that immediately after the cycle, which we
represent by a vector function Φ(N S , V S ) ≡ ((N S + n̄)/λ, φ(N S , V S )) defined in a spaceSS(N ) < V < Vsup
(N)}.
{(N, V )|K̄ < N < N̂ , Vinf

The discrete dynamics of Φ(·, ·) characterize the movements of cycles within thecyclical growth regime. The first element of the mapping, (N S + n̄)/λ, shows that after

many cycles N S will converge to a stationary point N ∗∗ = n̄/(λ − 1).42 By contrast, thefollowing shows that the movement of V S is unstable.

Claim 6 The slope of φ(N S , V S ) with respect to V S is larger than λ for all N S ∈ (K̄, N̂ )SS(Nt ), Vsup
(Nt )).
and V S ∈ (VinfNote also that, for any given N S , φ(N S , V S ) is a continuous function of V S and theS
S(N S ), Vsup
(N S )), is contained in its region (V̂ , (1−α)/ρN S ).
domain of this function, (Vinf

These facts guarantee that there is a unique stationary point of Φ(·, ·) and that this point
has a saddle property.

Let us denote the saddle point by (N ∗∗ , V ∗∗ ). Solving the discrete dynamics of Φ(·, ·)backwards from (N ∗∗ , V ∗∗ ), we can find the unique stable arm V S = SAcycle (N S ) that42

N ∗∗ is greater than the stationary point in the continuous growth regime, N ∗ .

− 28 −

satisfies
lim Φm (N S , SAcycle (N S )) = (N ∗∗ , V ∗∗ )m→∞for all N S ∈ [K̄/µ, N̂ ). Figure 5 depicts the stable arm and the movements of N S andV S . If a cycle starts from a state oﬀ the stable arm, the deviation of V S from the stable

arm is multiplied by more than λ > 1 in each cycle and the economy will enter the
infeasible region in finite cycles. Thus, provided that the economy stays in the cyclical
growth regime permanently, the initial value of V S is determined by SAcycle (N S ) andthe economy eventually converges to the stationary cycle, that is, the cycle starting from(N ∗∗ , V ∗∗ ). Combining the paths of N S and V S derived above with the dynamics of Nt
and Vt within a cycle, and from the fact that the frontier technology is multiplied by λ at
the beginning of each cycle, the movements of Nt , Vt and Zt in the cyclical growth regime
are uniquely determined. Then, from (14) and (17), the consumption of a representative
consumer follows
Dt =

5.1

αc(Nt − N S + K̄)(Zt Nt )(1−α)/α .
LVt

(39)

Movements of Real Aggregate Variables

Let ωt denote the real wage at time t, which also represents the real labor income of
the representative consumer since each consumer inelastically supplies one unit of labor
service. From the fact that the reciprocal of the price index coincides with the aggregate
consumption, which is given by (17), the real wage rate can be calculated as

Since Zt Nt representsRωt ≡ wt /P̄t = α(Zt Nt )(1−α)/α .

(40)

zdFt (z), which increases smoothly,43 real income always grows

monotonically and smoothly.
Dividing both sides of equation (14) by the price index, the real value of a frontier
firm is calculated as Vt /P̄t = ωt ltmin . This expression enables us to derive the movementof the real interest rate, denoted by Rt , within a cycle from the arbitrage condition inreal terms.
Rt =43

ω̇t l˙tmin 1 − α++.
ωt ltminN t VtUsing equations (8), (9) and (14), the time derivative of

α/vt (1).

R(41)

zdFt (z) is calculated as (Zt /ltmin )L −Here, Zt /ltmin = maxz z/c(Kt (z)) varies continuously due to the envelope theorem.

clear from equation (10) that vt (1) is continuous in t. Thus, the time derivative ofR
in t, which means that zdFt (z) changes smoothly.

− 29 −

R

It is also

zdFt (z) is continuousThe above equation shows that the real interest rate is the sum of the growth rate of
the real wage, that of R&D cost, and the reciprocal of the price-earning ratio, all ofwhich vary continuously within each cycle.44 However, as shown below, the real wage
drops discontinuously when a new cycle starts. Note that the sensitivity of R&D cost to
knowledge becomes discretely larger when the new technology is adopted, since there isa relatively small amount of knowledge about the new technology. This causes a sudden
spur in the reduction of the labor cost of R&D, which means a discrete reduction in thesecond term of (41).45 Thus, the real interest rate jumps downward when a new cycle
starts.
Given the streams of real income and the real interest rate, consumption by a representative consumer with a log utility function is determined according to Ḋt /Dt = Rt −ρ.

The time path of consumption is smooth within each cycle, but not on the points between cycles since Rt drops discontinuously when new technologies are adopted. Thisimplies that each cycle starts with a small, possibly negative, growth rate of consumption
relative to the average growth rate within the whole cycle. As a result, the time path of
consumption has a cyclical component.
Recall that consumption Dt also represents per capita production at each date. Using
real wage ωt , the total amount of production is expressed as LDt = α−1 ωt (L−LRt ), wheremin
L−LRt = αlt /Vt represents the number of production workers. This expression clarifies

that the initial decline in the growth rate of Dt is associated with a sudden increase in
the labor flow from production towards R&D. By adding the real value of new firmsRcreated through R&D, (vt (z)/P̄t ) dRt (z) = LRt ωt , to the amount of production, real

GDP is obtained,

∙

¸
1−α
R
(L − Lt ) + L ωt .
GDPt =
α(42)

Although the addition of R&D value reduces fluctuations in GDP relative to that in Dt ,GDP is still aﬀected by the distribution of the labor force between production and R&D.
Note that while R&D activities are competitive, the goods market is monopolistic and44

Since the time path of the real wage is smooth, its growth rate is continuous. The denominator of thethird term can be broken down into Nt Vt = (Nt Zt )vt (1). As shown in footnote 43, both Nt Zt and vt (1) arecontinuous in t, and thus the third term is continuous. Finally, the growth rate of ltmin = c(Nt − N S + K̄)
is also continuous in t within a cycle since Ṅt changes continuously according to (37) within a cycle.
45

Let denote Kt (Zt ) by k.

Then, l˙tmin /ltmin = (c0 (k)/c(k))(dNt /dt) = (c0 (k)/c(k)2 )LR=t−1
are decreasing in k as long as k ≥ K̄ and LR−²c (k)(c(k)k)−1 LR
t . Since both ²c (k) and (c(k)k)
t
min
min
does not jump, the absolute value of l˙t /lt jumps upward when k drops.

− 30 −

Figure 6: Movements of aggregate variables within three consecutive stationary cycles.
δ = 2, µ = 0.5.
there is a positive markup. Thus, an increase in the labor flow from production toward
R&D brings about a discontinuous fall in GDP growth, since it diverts resources awayfrom the monopolistic sector to the competitive sector.
Figure 6 shows a representative time path of the real wage (ωt ), consumption (Dt ),and per capita real GDP (GDPt /L), within a few consecutive stationary cycles, calculated numerically.46 All three variables have the same average growth rate ((1 −α)/α) log λ/T ∗∗ , where T ∗∗ represents the duration of each stationary cycle. In contrast

to the steady increase in the real wage, consumption has a large fluctuation including
negative growth at the beginning of each cycle and sustained growth only a few years
after the adoption of the new technology, though exact dynamics depend on parametervalues. The fluctuation in GDP is relatively small because of the inclusion of R&D value,
but its pattern is essentially the same as that of consumption.47
46

In all simulation results presented in this paper, we set parameter values to α = 0.5, ρ = 0.025, andL = 1, which are the same as those adopted in the numerical simulation in the HT model. We use a
simple R&D cost function c(k) = 1/k + 1 that satisfies assumption 1. Here, the two parameters of thespillover function are set to δ = 2 and µ = 0.5, though various values are examined later.
47

This kind of downturn in economic performance immediately after the adoption of new technology

has been referred to by some economists as productivity slowdown puzzle. See Greenwood and Yorukoglu− 31 −

5.2

Relationships with the literature of GPTs

The dynamics investigated in this section are characterized by dramatic innovationsin production technology and resulting discontinuities in the evolution of output andthe pattern of resource allocation. These outcomes are consistent with a number ofobservations made by economic historians in which major improvements in technology
had far-reaching and prolonged implications, such as the steam engine, electricity, andthe computer. To stylize these dramatic historical innovations, the notion of general
purpose technologies (GPTs) was recently introduced by Bresnahan and Trajtenberg(1995), and subsequently a number of theoretical research studies were carried out toinvestigate the relationship between GPTs and economic growth.
Although our model did not assume the existence of GPTs in its setup, it turns outthat each technology, chosen endogenously in the cyclical growth regime, has four characteristics which are necessary and suﬃcient for any technology to qualify as a GPT.48
First, a new technology has scope for improvement. Initially at the time of adoption thelabor cost of R&D is a high value of c(K̄), but it is continually reduced until c(K̄)/λat the end of cycle.49 Second, each technology is used to produce a wide variety of
products. Throughout each cycle, all entrepreneurs adopt the same technology to produce distinct products for diﬀerent purposes. Third, the range of the new technology,
defined as the proportion of productive activities in the economy using that technology,
becomes considerably wider within the cycle. Specifically, the share of production basedon the current frontier technology is calculated as (Nt − N S )/Nt , which initially starts

at zero and continuously increases until it reaches the maximum value n̄/(N S + n̄) at
the end of the cycle.50 After that, the share of technology z gradually declines according
to zn̄/Zt Nt .51 Fourth, there are strong technological complementarities between R&D
activities through spillovers of knowledge. Due to the spillover eﬀect within the current(1994) and Hornstein and Krusell (1996). Note also that our model predicts low real interest rates during
the periods of slowdown.
48

These criteria were presented by Lipsey, Bekar and Carlaw (1998).

49

Our specification limits the improvement to the R&D cost because the number of firms is normalized

so that their products enter the utility function symmetrically. Nonetheless, it is possible to reinterpret
this cost reduction as an improvement in the quality of products by adopting another measure of firms.
50
51

In a stationary cycle, this maximum value is (λ − 1)/λ, which is reasonably large.
On this point, the workings of our model are more realistic than the dynamics of the HT model,

where only one GPT is used in production at each date.

− 32 −

frontier technology, adoption of that technology by some firm makes it more attractive forsubsequent entrepreneurs to adopt the same technology. Moreover, there exist spillovers
across technologies, which enable the economy to shift to more advanced technology
when the scope of cost reduction in the current technology has largely been exploited.
One major contribution of this paper is that it endogenizes the process by which GPTs
emerge from a continuum of potentially available technologies. While existing models
take the time of arrival of new GPTs and/or the productivity of GPTs as exogenous, ourmodel explains the way in which both are determined. In addition, it shows that growthwith GPTs is not the only possibility: as investigated in the previous section, there isanother regime of economic growth where new technologies are continuously adopted
and any single technology does not have a positive share (hence it is not a GPT). Thus,in our framework, it is possible to examine whether or not the formation of GPTs isbeneficial for the economy. The next section deals with this issue.

6

WelfareIn the previous sections we have derived two patterns of equilibrium dynamics in the period of sustained growth, respectively called the continuous and cyclical growth regimes.52
In a given economy, both patterns of dynamics may be realized and, moreover, it is possible that the economy switches from one regime to another at certain points in theequilibrium path.53 This implies that there are many perfect foresight equilibria, each ofwhich diﬀers in the pattern of technological adoption. The pattern of technological adoption aﬀects the evolution of consumption of a representative consumer, and consequently
52

The dynamics in these two regimes are the only equilibrium dynamics that satisfy the condition

that no entrepreneur adopts technologies that are behind the current frontier. Any equilibrium path notsatisfying this condition is unstable because such an equilibrium path must contain a finite time period
in which multiple technologies are adopted. Consequently, it has a ’knife-edge’ property in that, oncethe distribution of R&D over these technologies is disturbed, these multiple technologies cannot havethe same profitability over that time period and thus only one technology will be adopted. Unless all
entrepreneurs are able to coordinate their pattern of technology adoption, such a path is unlikely to berealized.
53

An economy can switch to the cyclical growth regime from any point in the continuous growth regime.

By contrast, when an economy is in the cyclical growth regime, it can switch to the continuous growth
regime only after the current cycle is finished (that is, only after the frontier jumps to a new technology).
It is possible to show that once the pattern of switching between the two regimes is specified, a uniqueperfect foresight equilibrium exists that follows that pattern.

− 33 −

consumer welfare is also aﬀected.
This section investigates whether discrete technological progress with GPTs is moredesirable than continuous technological progress, in terms of the welfare of the representative consumer. First, we calculate the utility of the consumer in each regime, partlywith the aid of a computer, and show that the desirable pattern of technology adoption
diﬀers depending on parameters. This result will then be interpreted by showing that
there are both advantages and disadvantages in forming GPTs.
On the balanced growth path (BGP) of the continuous growth regime, Nt and Vt stayat their stationary values, N ∗ and V ∗ , respectively, and the technology, Zt , progresses
at a constant rate of g ∗ . Applying these facts for (35) and then substituting this into
(3), we obtain the utility of the representative consumer on the BGP of the continuous
growth regime.
∙
µ
¶¸
1
g∗
αc(K̄) 1 − α
1−α∗+U (Zt ) =ln
ln N +
+
Zt .
∗
ρ
LV
α
ρ
αρ
∗

On the other hand, state variables are cyclically evolving in the cyclical growth regime,
even after both N S and V S settle to their stationary values. Let Nτ and Vτ (0 ≤ τ ≤T ∗∗ ) denote the evolution of the state variables within the stationary cycle,54 whichis uniquely determined by (37) and (20) along with initial conditions N ∗∗ and V ∗∗ .
Then, substituting these paths into (39) yields the movements of consumption within(1−α)/αthe stationary cycle, which can be denoted by Zt

Dτ∗∗ . While the technology, Zt , isunchanged during one cycle, it is multiplied by λ when the economy enters the new cycle,
implying that the whole path of consumption in the new cycle is λ(1−α)/α times larger
than the last one. From these considerations, we can derive the utility level measured at
the starting point of a cycle:
1
U ∗∗ (Zt ) =∗∗1 − e−ρT

"Z

0

T ∗∗

ln Dτ∗∗ dτ + e−ρT ∗∗#1−α1−αln λ +Zt .
αραρ

The above two expressions give the utility of the representative consumer in thesteady state of each regime, but they are not suﬃcient to judge which regime is betterin a welfare sense since it is not possible to switch from the steady state of one regime
to that of the other immediately. Suppose that the economy is currently on the BGPof the continuous growth regime and then switches permanently to the cyclical growth54

Here index τ is used instead of t to express the time passed after the current cycle started. Recall

that T ∗∗ represents the time length of the stationary cycle.

− 34 −

Figure 7: Comparison of welfare between the two regimes.
regime. In this case, the first cycle in the cyclical growth regime starts from N S = N ∗and then, following the stable arm of Φ(·, ·) shown in figure 5, the economy graduallyconverges to the stationary cycle, where N S = N ∗∗ . Likewise, if the economy switchesfrom the stationary cycle to the continuous growth regime permanently, the transitional
dynamics start from Nt = N ∗∗ and gradually converge to the BGP, where Nt = N ∗ ,following the stable arm depicted in figure 2.
For various values for parameters,55 we have numerically calculated the utility of
the representative consumer when the economy switches from the steady state of one
regime to the other regime, and compared them with U ∗ (Zt ) and U ∗∗ (Zt ). The results
are summarized in figure 7, which shows that there are three regions in the relevant
parameter space. When the parameters belong to the lower region labeled ‘Cyclical’
the cyclical growth regime is desirable regardless of the initial state of the economy.56
Alternatively, when the parameters are in the upper region labeled ‘Continuous’ the55

In this simulation, we have focused on δ and µ, the parameters of the spillover function, since themost important diﬀerence between the two regimes is in the pattern of knowledge accumulation which
depends on these parameters. For other parameters, see footnote 46.
56

That is, the utility of remaining in the stationary cycle is larger than the utility gained from switchingfrom it to the continuous growth regime, and the utility of staying at the BGP is smaller than the utility
gained from switching from it to the cyclical growth regime.

− 35 −

continuous growth regime is desirable. In the intermediate region labeled ‘Stay’, it isbetter to stay in the current regime than to incur welfare losses during the transition.
This mixed result implies that there are both advantages and disadvantages in theformation of GPTs compared to shifting continuously to the advanced technologies. The
magnitudes of these eﬀects are influenced by the parameters of the spillover function,
δ and µ. As we have seen in the previous section, when a GPT is formed the cost of
R&D is eﬃciently reduced through the utilization of the technology-specific portion ofknowledge. In fact, when µ is smaller than unity (i.e., there is a positive portion ofknowledge that cannot be applied to other technologies), the growth rate of Zt Nt islarger in the cyclical growth regime, at least at the beginning of a cycle, than in thecontinuous growth regime for a given schedule of R&D labor input.57 Thus, for sometime after a cycle starts, the formation of a GPT promotes growth measured in termsof Nt Vt , which is beneficial for consumers since the per capita consumption is given by(Zt Nt )(1−α)/α times the portion of production labor. This eﬀect is stronger when µ is
smaller, which is consistent with the configuration in figure 7.
However, in the long term, GPTs may harm growth. Once a GPT is formed, it
creates a discrete diﬀerence between the cost of R&D based on the GPT and those basedon other technologies. Consequently, entrepreneurs are not willing to switch to advanced
technologies until a technology with a suﬃciently high productivity becomes available
at a certain cost of adoption. Thus, the productivity of the next technology must be
significantly higher than that of the current GPT. However, this means the spilloverof knowledge from the experiences in the current GPT to the next technology is small
in magnitude, especially when δ is large.58 In such a case, the adoption cost of newtechnology declines only slowly. This makes the economy stick to the current GPT evenafter the cost of R&D based on this technology becomes insensitive to the additionalamount of knowledge. By contrast, in the continuous growth regime, spillovers between
adjacent technologies promote adoption of advanced technologies and firms do not stick
to any technology. In fact, unless δ is close to unity, the cumulative amount of R&D
57

Let At ≡Rt
0LR
t dt denote the cumulative input of R&D labor. Then, at the beginning of a newcycle, d(Zt Nt )/dA = Zt /c(K̄) and d2 (Zt Nt )/dA2 = Zt /δ K̄c(K̄)2 . In the continuous growth regime,d(Zt Nt )/dA = Zt /c(K̄) and d2 (Zt Nt )/dA2 = µZt /δ K̄c(K̄)2 . While the first derivatives are the same,the second derivative for the case of cyclical growth is larger than that for continuous growth, as long asµ < 1.
58

Note that a large δ means that the magnitude of spillover is highly sensitive to the diﬀerence in the

productivity of technologies.

− 36 −

labor input that is required for the economy to switch to the next technology in thecyclical growth regime is larger than the amount required to reach the same technology
in the continuous growth regime. This implies that, in the long run, the average rate of
growth tends to be larger in the continuous growth regime than in the cyclical growth
regime.
The relative gap in the average rate of growth between the regimes increases with
δ, but this does not mean that the welfare disadvantages of the formation of a GPT
monotonically increase with δ. As δ gets larger, the growth rate itself declines in bothregimes. In addition, in the cyclical growth regime the length of one cycle becomes
longer. When δ is very large, the delay in the adoption of the next technology caused by
the formation of a GPT has only a minor eﬀect on overall welfare, since consumption insuch a distant future is heavily discounted. In fact, figure 7 shows that not only when δ issmall but also when it is very large, the cyclical growth regime is more desirable than thecontinuous growth regime, while the disadvantages of a GPT outweighs its advantages
when δ is an intermediate value (provided that µ is not small).

7

Conclusion

We have demonstrated that when there is a continuum of technologies diﬀering in productivity, the pattern of growth is not unique. When the portion of technology-specificknowledge is large, it is desirable to form a sequence of GPTs. These GPTs produce
cyclical evolutions in the macroeconomic variables, including a slowdown in GDP growth
immediately after the economy switches to a new technology. On the other hand, in an
economy where the portion of knowledge that can be applied to adjacent technologies is
not quite so small, but it is reasonably diﬃcult to apply knowledge to other technologies
that diﬀer substantially in productivity, shifting continuously to advanced technologies
results in higher welfare than forming GPTs. In this case, all the macroeconomic variables are evolving smoothly, converging to the balanced growth path where GDP grows
at a constant rate.
It is not guaranteed, however, that technologies are adopted in the desirable pattern since both patterns (and their possible hybrids) always constitute perfect foresightequilibria and it depends on the expectations of entrepreneurs which pattern is realized.59 This suggest the possibility that some policies may improve the welfare of the59

In reality, it may even possible that agents have diﬀerent expectations, though this viorate the− 37 −

representative consumer by aﬀecting the pattern of technological adoption. Specifically,
with various measures to promote the adoption of specific technologies, such as direct
subsidies or public research to aid the accumulation of knowledge, the government canmanipulate the relative profitability of adopting various technologies, lt (·), and lead theeconomy to the desirable regime.60 In general, even when the economy is currently in thedesirable regime, there remains room for government intervention since R&D has positive
externalities. While this property is common to many endogenous growth models, ourmodel suggests that policies must be implemented in a careful way: if the possibility ofa regime change is not considered, some policies that are intended to improve welfare,assuming that the economy stays in the current regime, may worsen the situation byleading it to the Pareto inferior regime.

Appendix A: Proof of Claims
A.1

Proof of Claim 1

Let us define a function g(x) ≡ ln s(1, ex ). Note that s(z0 , z) = s(1, z/z0 ) holds sinces(·, ·) is homogeneous of degree 0. Then, for any given ζ ≡ ln(z/z0 ), property d ofassumption 2 implies that g(ξ) + g(ζ − ξ) does not depend on ξ ≡ ln(z 0 /z0 ) ∈ (0, ζ).

From this, it is clear that the derivative of g(ξ) + g(ζ − ξ) with respect to ζ, g 0 (ζ − ξ),does not depend on ξ ∈ (0, ζ). Since ζ can be any positive number, the result is thatg 0 (x) is constant for all x > 0. Thus, for some constants c0 and c1 , g(x) can be expressed

as c0 + c1 x. Then, we obtain s(z0 , z) = eg(ln(z/z0 )) = ec0 +c1 ln(z/z0 ) = ec0 (z/z0 )c1 for any
conventional assumption of rational expectations. For example, suppose that there are two groups ofentrepreneurs. One group expects that the economy will follow the continuous growth regime, while
the other expects the cyclical growth regime to occur. Each group carries out R&Ds according to theirbelief. Depending on which belief is held, the discounted sum of profits that can be gained from investingin R&D, Vt , diﬀers. In the labor market, the group of entrepreneurs who believe Vt is larger than theother group believes, hires labor for R&D until the wage level rises to satisfy the free entry condition
under their belief. Since the other entrepreneurs believe that their investments will not pay at this wage
level, only those entrepreneurs who expect large Vt invest in R&D and, as a result, their expectations
are actually realized. The outcome is essentially the same as the perfect foresight equilibrium where all
entrepreneurs expect the regime that realizes large Vt . However, since the high value of Vt means thatthe goods market in the future is less competitive, it tends not to be the desirable equilibrium in termsof welfare.
60

To this end, the scale of intervention (such as the amounts of subsidies) can be arbitrarily small,which implies that a marginal intervention can improve welfare discretely.

− 38 −

z > z0 and s(z0 , z) = ec0 (z0 /z)c1 for z < z0 from the assumption of symmetry. For thefunction s(·, ·) to be bounded in (0, 1], ec0 must be bounded in (0, 1]. From the propertye, c1 must be smaller than −1. This completes the proof.

A.2

Proof of Claim 2

We first show that Lminis bounded. Let us introduce a function ˜lt (ζ) ≡ ln c(Kt (Zt )e−δζ /µ)−t

ζ, which is no larger than ln lt (Zt eζ ) = ln c(Kt (Zt eζ )) − ζ since Kt (Zt )(z/Zt )δ /µ ≥ Kt (z)holds for all z. Its derivative is ˜l0 (ζ) = δ²c (Kt (Zt )e−δζ /µ) − 1. From assumption 1,t

we have limζ→∞ ˜lt0 (ζ) > 0 and limζ→−∞ ˜lt0 (ζ) < 0. Then, it follows that limz→∞ lt (z) ≥limζ→∞ exp(˜lt (ζ)) = ∞ and limz→+0 lt (z) ≥ limζ→−∞ exp(˜lt (ζ)) = ∞. Thus, there exists/ [z, z̄].
a closed interval [z, z̄] such that lt (z) > lt (Zt ) for all z ∈Next, we examine the existence of ltmin . If µ = 1, the existence is obvious since lt (z)is a continuous function. In the following, we focus on the case µ < 1. Let Mt ≡ {z ∈[z, z̄]|Ft (z) − Ft (z−) > 0} denote the set of all mass points in [z, z̄], and denote by M̄t

the complementary set [z, z̄]\Mt . Since lt (z) is not continuous on Mt , let us introduce

a continuous variant of lt (z), lt− (z) ≡ (Zt /z)c(Kt− (z)), whereKt− (z) = µ

Z ∞
0

½ 0
¾
z z δ
min
dFt (z 0 ).
,
z z0

(A.2)

Note that lt− (z) is larger than lt (z) on Mt and coincides with lt (z) on M̄t . Sincelt− (z) is continuous and [z, z̄] is compact, there exists lt− min ≡ minz∈[z,z̄] lt− (z).

Ifminz∈Mt lt (z) exists, ltmin is the smaller of this and lt− min . If minz∈Mt lt (z) does notexists but inf z∈Mt lt (z) is no less than lt− min , then ltmin = lt− min . Finally, suppose thatminz∈Mt lt (z) is non-existent and that inf z∈Mt lt (z) is smaller than lt− min . Then, we canchoose some ξ > 1 so that there are infinite elements in Mξt ≡ {z ∈ Mt |ξlt (z) <lt− min }. For each z ∈ Mtξ , Ft (z) − Ft (z−) ≥ k ln ξ/(1 − µ)²c (k) > 0, where k ≡R∞PµKt (Zt ) min{z/Zt , Zt /z̄}. Thus 0 zdFt (z) ≥ z z∈Mξ (Ft (z) − Ft (z−)) = ∞, whichtcontradicts assumption 4.

A.3

Proof of Claim 3

R∞
From equation (7) and (10) and the fact that Zt Nt ≡ 0 z dFt (z) is nondecreasing inR∞time, it follows that Vt = (1 − α)Zt t e−ρ(τ −t) (Zτ Nτ )−1 dτ ≤ (1 − α)/ρNt . Note thatthe equality in the latter applies if and only if Lτ = 0 holds for all τ ≥ t, since otherwiseZτ Nτ increases at some date in the future. On the other hand, we have shown in the− 39 −

text that as long as there is a positive amount of R&D, Vt = wt ltmin and wt > α/L hold,which implies that Vt > αltmin /L. This completes the proof of the part a.
It is clear from the above argument that when αltmin /L is larger than or equal to(1 − α)/ρNt there will be no R&D carried out in the economy. Conversely, consider thecase where (1 − α)/ρNt is larger than αltmin /L. Suppose that there is no R&D at timet. In this case, the free entry condition requires Vt ≤ αltmin /L since no R&D implies

wt = α/L. If there is no R&D for all dates in the future, Vt = (1 − α)/ρNt > αltmin /L
holds, which contradicts the free entry condition. Thus, some entrepreneurs must carryout R&D in the future. Let t0 be the first date after t with LR
t0 > 0. Then, the part a/L. Since there are no movements in the distributionof the claim requires Vt0 > αltmin
0= ltmin and Zt0 = Zt hold.
of firms within the time interval between t and t0 , ltmin0/L < Vt0 =These facts and the non-decreasing nature of Zt Nt imply αltmin /L = αltmin0
R ∞ −(τ −t0 )R ∞ −(τ −t0 )Zτ Nτ dτ ≤ (1−α)Zt t0 eZτ −t0 +t Nτ −t0 +t dτ = Vt , which again(1−α)Zt0 t0 e
contradicts the free entry condition at date t. Therefore, whenever (1 − α)/ρNt >αltmin /L, there must be a positive amount of R&D in the economy.

A.4

Proof of Claim 4

Let Kt− (z) be defined by equation (A.2). We first derive the slope of Kt− (z). Its left-handderivative is calculated as(µ ¶µ 0 ¶δ )zz0 δ−
dFt (z 0 )zz−²(0,z−²]
(µ ¶
µ
¶ )
Z
1
z−² δ
z0 δ
+ µ lim−dFt (z 0 )²→+0 ² (z−²,z)
z
z0
(
)
Z
³ z ´δ µ z − ² ¶δ
1
+ µ lim−dFt (z 0 )²→+0 ² [z,∞)z0z0K − (z) − Kt− (z − ²)1=µ limlim t²→+0²→+0 ²²

Z

The absolute value of the second term is bounded by
¯µ ¶
µ
¶ ¯
Z
Z
1 ¯¯ z 0 δ
z − ² δ ¯¯
2µδ
0
dF
−
(z
)
=
dFt (z 0 ),
lim
µ lim
¯
¯ t
²→+0 (z−²,z) ² ¯ z¯
z0z ²→+0 (z−²,z)which is zero. Thus, the left-hand derivative of Kt− (z) is calculated as#" Zµ 0 ¶δ
Z³ z ´δzµδ
dFt (z 0 ) +dFt (z 0 ) .
−
0
z
z
(0,z)
[z,∞) z− 40 −(A.3)

In a similar way, we can derive the right-hand derivative of Kt− (z) as#" Zµ 0 ¶δ
Z³ z ´δµδz
dFt (z 0 ) +dFt (z 0 ) .
−
0
z
z
(0,z]
(z,∞) z

(A.4)

Note that the two expressions, (A.3) and (A.4), coincide when Ft (z) − Ft (z−) = 0,

that is when z is not a mass point of Ft (·). In this case, Kt− (·) is diﬀerentiable at z andits derivative is unambiguously expressed as#" Z µ ¶
Z ∞³ ´
z
δµ
z0 δ
z δ
−0
0
0
dFt (z ) +dFt (z ) .
−Kt (z) =
z
z
z0
0
z

(A.5)

The assumption Zt0 < z for all t0 < t means that z is not a mass point and that0

the second term in the square bracket in (A.5) is zero. In this case, Kt− (z) reducesto −(δ/z)Kt (z).

From this result, we can also calculate the derivative of lt− (z) ≡0(Zt /z)c(Kt− (z)) as lt− (z) = (lt− (z)/z) (δ²c (Kt (z)) − 1) , which becomes zero if and only
0−0
if Kt (z) = ²−1c (1/δ) ≡ K̄. If lt (z) were not zero, there would exist some technology z

such that lt− (z 0 ) is smaller than lt− (z). In this case, lt (z) = lt− (z) > lt− (z 0 ) ≥ lt (z 0 ) holds

and thus z is not contained in Lmint , contradicting (15).

A.5

Proof of Claim 5

a. Note that d ln c(k)/d ln k = −²c (k) > −1/δ = d ln c̃(k)/d ln k holds for all k > K̄. Sincec(·) is a decreasing function, ln c(K̄/µ) is smaller than ln c(K̄) = ln c̃(K̄/µ). Let K̃ ≡(K̄/µ) (c(K̄)/c(K̄/µ))1/(1/δ−²c (K̄/µ)) > K̄/µ. Then, ln c(K̃) = ln c((K̄/µ) exp((ln c(K̄) −ln(K̄/µ))/(1/δ − ²c (K̄/µ)))) is larger than ln c(K̄/µ) − (1/δ)(ln c(K̄) − ln(K̄/µ))/ (1/δ −²c (K̄/µ)) = ln c̃(K̃). It follows these facts that ln c(k) intersects with ln c̃(k) from below
only once in the region k > K̄ and the point of intersection K max is located between
K̄/µ and K̃.
0b. Suppose that a technology z 0 ≤ z exists such that z 0 ∈ Lmint . Then lt (z ) must

be no greater than lt (z), which requires Kt (z 0 ) ≥ Kt (z) ≥ K max . Let us take another technology z 00 = (µKt (z 0 )/K̄)1/δ z 0 . Then, Kt (z 00 ) ≥ µKt (z 0 )(z 00 /z 0 )−δ = K̄. Fromthese properties, it follows that lt (z 00 ) ≤ (Zt /z 00 )c(K̄) = (Zt /z 0 )(K̄/µKt (z 0 ))1/δ c(K̄) =/ Lmin(Zt /z 0 )c̃(Kt (z 0 )) < (Zt /z 0 )c(Kt (z 0 )) = lt (z 0 ), which implies z 0 ∈t , a contradiction.

⊂ (z, ∞) follows.
Thus, Rt (·) ⊆ Lmintand Ft (z 0 ) − Ft (z 0 −) =c. Suppose that a technology z 0 ≤ z exists such that z 0 ∈ Lmint

0. Then lt (z 0 ) must be no greater than lt (z), which requires Kt (z 0 ) ≥ Kt (z) > K̄. Since− 41 −

z 0 is not a mass point, Kt (z 0 ) coincides with Kt− (z 0 ), which is defined by (A.2). The
right-hand derivative of Kt− (·) at z 0 is derived in a similar way to the way we havederived (A.4), which turns out to be no smaller than −(δ/z 0 )Kt− (z 0 ). From this, itfollows that the right-hand derivative of lt− (·) at z 0 is well defined and no greater than¡¢(lt− (z 0 )/z 0 ) δ²c (Kt− (z 0 )) − 1 , which is strictly negative because of Kt− (z 0 ) > K̄. Thismeans that there exists some technology z 00 > z 0 such that lt− (z 00 ) < lt− (z 0 ). Sincelt (z 0 ) = lt− (z 0 ) > lt− (z 00 ) ≥ lt (z 00 ), z 0 is not contained in Lmint , contradicting the initialassumption of z 0 ∈ Lmint .

d. When µ = 1, the definition of Kt− (·) coincides with that of Kt (·). Thus, the proof

presented in part c applies regardless of whether z 0 is a mass point or not.

A.6

Proof of Claim 6

Consider a possible equilibrium path in one cycle that starts from (N S , V S ), whereS
S(Nt ), Vsup
(Nt )). Then, at each point, the slope of the path
N S ∈ (K̄, N̂ ) and V S ∈ (Vinfin (Nt , Vt ) space is V̇t /Ṅt , where Ṅt and V̇t are given by (37) and (20), respectively. Notethat this slope is strictly negative because Ṅt > 0 and V̇t < 0. Let V E denote the valueof Vt when Nt reaches N S + n̄.
00Suppose that there is another possible path that starts from (N S , V S ), where V S ∈S(Nt )). Consequently, this path must be located above the previous one at each
(V S , Vsup

Nt ∈ [N S , N S + n̄]. Moreover, the slope of this path at each Nt is larger than than thatof the previous path since

d
dVt

Ã

V̇t
Ṅt

!

1
=
Ṅt

Ã

α V̇t
ρ+ 2
Vt Ṅt

!

>0

(A.6)

holds for all Nt ∈ [N S , N S + n̄]. This means that the gap between the two paths gets
00

wider as Nt increases, which implies V E − V E > V S − V S . After the frontier jumps,000

the gap widens: φ(N S , V S ) − φ(N S , V S ) = λV E − λV E > λ(V S − V S ).
0

Note that the opposite inequality holds when V S < V S . Either way, we finally obtain
0

φ(N S , V S ) − φ(N S , V S )>λ
V S0 − V S
0for all V S 6= V S , which completes the proof.

− 42 −(A.7)

Appendix B: Definition of Γ
max the solution to c(K max ) = c(K̄)(K̄/µK max )1/δ in domain
Let K̄ denote ²−1c (1/δ), K(K̄/µ, ∞), and λ the solution to c(K̄(λδ − 1 + µ)/µ) = c(K̄)/λ in domain (1, ∞), themeanings of which are explained in claim 4, claim 5 and section 5, respectively. Then,
¾
½
δ
max
max K̄c(K̄)(λ − 1)c(K),.
Γ = max Kµ(λ − 1)

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