GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.282

Analysis of Transition Dynamics caused by Technological
Breakthroughs-Cause of productivity slowdown and drop in existing firms' stock prices-

Kazuyoshi Ohki

October 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Analysis of Transition Dynamics caused by Technological
Breakthroughs∗-Cause of productivity slowdown and drop in existing firms’ stock pricesKazuyoshi Ohki†
2012

Abstract
This paper examines the transition dynamics caused by technological breakthroughs. Our results show thattechnological breakthroughs cause a productivity slowdown and a drop in the stock prices of existing firms; thesefindings are consistent with observations in the 1970s. We explain how technological breakthroughs cause these
phenomena. The emergence of a new technology creates new business opportunities, which reduces existing
firms’ profits, thus causing their stock prices to drop. This decline in existing firms’ profits discourages R&Dintensive firms from entering the sector; this decreases aggregate R&D activity, and thus the growth rate ofproductivity declines.

keyword: R&D, technological breakthrough, productivity slowdown, stock price, general purpose technology

JEL classification: O11, O31, O41∗ I thank Koichi, Futagami. Shinsuke, Ikeda. Tatsurou, Iwaisako. Takeshi, Ojima. and participants at Japanese Economic Association Autumn meeting in Kyushu Sangyo University for their helpful suggestions. Any remaining errors are my responsibility.
† Graduate
School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan.
emai:kazuyoshi.ohki@gmail.com

1

1

Introduction

Since the beginning of the Industrial Revolution in Britain (1760–1850), we have witnessed a number of dramatic
technological breakthroughs, which have had enormous and long-term impacts on economies. Schumpeter refers
to these as Kondratieﬀ cycles, caused by technological factors. For example, electrical technologies had a major
inﬂuence between 1890 and 1930, and the impact of information technology (IT) during the second half of the 20th
century was also important. These technologies created new markets that opened up new business opportunities,
and induced a sequence of secondary incremental innovation, which Bresnahan and Trajtenberg (1995) characterized
as general purpose technology (GPT). For example, household electrical appliance manufacturers and electric powercompanies both arose as a consequence of electricity, and now continue to improve the quality of their products tosustain consumer demand. Similarly, PC manufacturers, software companies and Internet service providers were
set up following the emergence of IT. These companies also continually improve the quality of their products.
The purpose of this paper is to analyze the transition from the emergence of a new technology to its maturity.
In particular, we consider the growth rate of productivity, the stock price of incumbents in the preexisting sector
and aggregate stock price. Jovanovic and Rousseau (2005) present evidence that, following the emergence of a newand transforming technology, the growth rate of productivity declines for some decades, after which it begins to riseagain. Indeed, the productivity slowdown is now recognized as a stylized fact. Our model describes these changes
and shows that the slowdown is caused by declining R&D in existing technology. Greenwood and Jovanovic (1999)
present evidence that, upon the arrival of a new technology, existing ﬁrms’ stock prices fall, never to recover. They
also show that several years after the emergence of a new technology, aggregate stock prices rise because of the riseof the prices of the stock of new ﬁrms exploiting the new technology. Our model describes these facts and showsthat the decline in the stock prices of existing ﬁrms is caused by the decline in their proﬁts, while the aggregate
stock price rises with the new sector’s maturity.
The model is the extension of Segerstrom (1998), which is a quality-ladder R&D model, freed from scale eﬀects
by assuming that the diﬃculty of R&D increases as R&D accumulates.1 His model is a semi-endogenous growthmodel, where the long-run growth rate is never inﬂuenced by political variables or preferences. More recently, theshort-run eﬀects in semi-endogenous growth models have received more attention. Steger (2003) found that the
speed of convergence in Segerstrom (1998) was only 0.019, which means that it takes about 38 years for half theinitial gap to vanish (this is a very long time!). Therefore, analyzing the transition dynamics is very important.
In this paper, we examine the technological cycle by analyzing the transition dynamics of an extended Segerstrom
1 Aghion and Howitt (1992),Grossman and Helpman (1991) and Romer (1990) pioneered these R&D models.

2

(1998) model, which has very long transition dynamics.
In this paper, we expand Segerstrom (1998) by introducing heterogeneity across sectors in four ways: eﬃciency,diﬃculty and cost of R&D, and market size.2 This extension enables the model to simultaneously analyze boththe “old sector” and the “new sector”, where the old sector has low R&D costs and a high diﬃculty of R&D , andthe new sector has high R&D costs but a low diﬃculty of R&D. We examine the process caused by a technologicalbreakthrough to analyze the transition dynamics from the existing old-sector equilibrium alone to the equilibriumof both the old and new sectors.
Consider the following situation. At stage 1, there exists an “old sector” with a continuum of products. At
stage 2, an exogenous technological breakthrough occurs, leading to emergence of a “new sector” with a continuumof products. The initial cost of R&D in the new sector is high enough that no R&D is conducted in this new sector.
Stage 3 is reached later, when another exogenous shock occurs, which drastically reduces the cost of R&D in the
new sector, such that it is now conducted. This paper examines how economically important variables such as thegrowth rate of productivity and stock prices respond to these two enormous exogenous shocks.3
Some stage 2 results are consistent with empirical facts. First, the growth rate of productivity declines. New
technology creates new business opportunities, thus forming the new sector, which captures market share from oldsector ﬁrms, thus causing their proﬁts to decrease. This decreases the incentive to enter the old sector, reducing
R&D in the old sector. Thus, the instantaneous probability of successful R&D in the old sector decreases. On
the other hand, R&D is not being conducted in the new sector because of the very high cost of R&D. Therefore,
aggregate R&D decreases, which leads to a decrease in the aggregate probability of successful R&D. Because thegrowth rate of productivity is calculated as the weighted sum of probabilities of successful R&D, this reduces
productivity growth. Second, old-sector incumbents’ stock prices decrease because their discounted stream of
proﬁts has decreased.
Stage 3 results are also consistent with empirical facts. First, the growth rate of productivity rises. Several years
after a new technology is born, the cost of R&D falls. R&D activity then begins in the new sector, which dominates
the eﬀect of the decline of R&D activity conducted in the old sector. Therefore, aggregate R&D increases, whichleads to an increase in the aggregate probability of successful R&D, and the growth rate of productivity rises.
Second, old-sector incumbents’ stock prices remain low. Because old-sector market share does not change between
stages 2 and 3, the discounted proﬁts of old-sector incumbents remain small and hence their stock prices remain
2 Smulders and van de Klundert (1995), Startz (1998), Li (2000), Boucekkine and de la Croix (2003), Doi and Mino (2005) and Chu(2011) have investigated growth models with multiple sectors.
3 By assuming that R&D cost decreases due to learning by doing, we can extend the model so that the time of initial new-sector R&D
investment is determined endogenously; however, including these would complicate the model signiﬁcantly and so is left for further
research.

3

low. Third, the aggregate stock price rises when we include the new-sector ﬁrms, which were created in responseto the new technology. As the new technology matures, the status of new-sector ﬁrms stabilizes, raising theirdiscounted proﬁts. Hence, new-sector ﬁrms’ stock prices rise, dominating the decline of old-sector incumbents’stock prices. Therefore, the aggregate stock price rises.
The contributions of this paper are as follows. First, we explain the transition of economically importantvariables such as productivity and stock price that is observed empirically using a typical R&D growth model.
Second, we construct a quality-ladder R&D model with ex ante heterogeneity (eﬃciency, diﬃculty and cost of R&D,
and market share diﬀerences across sectors evaluated before R&D is conducted). The current literature examines
quality-ladder R&D models with ex post heterogeneity (eﬃciency of product technology, degree of increment quality,
shadow price and proﬁts diﬀer across ﬁrms evaluated after having conducted R&D).4 Our model extends thesemodels in another direction. Third, we carefully analyze the transition dynamics of a semi-endogenous growthmodel. As discussed in Steger (2003), the transition path of a semi-endogenous growth model is so long thatanalyzing transition dynamics is important for development of this model.

1.1

Related literature

Helpman and Trajtenberg (1998) and Aghion and Howitt (2008) analyze the relationship between the emergenceof a new technology (GPT) and productivity slowdown. Helpman and Trajtenberg (1998) state that because
labor is needed to introduce a new GPT, the amount of labor engaging in manufacturing declines, thus reducing
output. Aghion and Howitt (2008) use a simple neoclassical model to consider a rate of capital obsolescence thatis proportional to the innovation rate. They state that a new GPT raises both the innovation rate and the rate ofcapital obsolescence, which reduces the growth rate. On the other hand, this paper states that a new technology
creates a new sector and diminishes the incentive to carry out R&D in the old sector, which reduces the growthrate of productivity.
Greenwood and Jovanovic (1999) and Hobijn and Jovanovic (2001) analyze the relationship between the emergence of IT and incumbents’ stock prices. They use the Lucas tree model and assume that the emergence of IT
causes permanently higher income several years later. The results show that the substitution eﬀect causes current
consumption to rise, and thus this reduces demand for assets (the right to future consumption), which reduces
asset prices. On the other hand, we show that the initial fall in the stock price is due to the decline in old-sector
incumbents’ proﬁts and that the subsequent rise is due to the maturation of the new technology sector.

4 Dinopoulos and Unel (2011) and Minniti, Parello and Segerstrom (2012) examine R&D with ex post heterogeneity.

4Cheng and Dinopoulos (1996) analyze the endogenous deterministic cycles caused by technological breakthroughs. However, they do not examine the eﬀects of technological breakthroughs on the dynamics of productivity
or stock price growth rates. Boucekkine and de la Croix (2003) study the eﬀect on the growth rate of the IT revolution, which changes the productivity of the ﬁnal goods sector and the eﬃciency of the R&D sector. They use asimulation model in which eﬃciency of R&D increases permanently and productivity of the ﬁnal goods sector decreases temporarily to show that productivity slowdown occurs. They state that the productivity of labor initiallydecreases because of the negative shock to the ﬁnal goods sector. However, the positive shock to the R&D sector
stimulates R&D activity, which increases productivity. Thus, the productivity of labor eventually increases. Thismechanism is clearly diﬀerent from our model.
The rest of this paper is organized as follows. Section 2 describes the basic setup of the model. Section
3 describes the path of the economy and establishes the balanced-growth equilibrium. Section 4 analyzes thetransition from the emergence of the new technology to its maturity. Section 5 concludes.

2

The Model

2.1

Environment

The economy consists of three agents: households, R&D ﬁrms and producing ﬁrms (incumbents). Households
supply labor inelastically, save by investing in R&D ﬁrms as stockholders and consume products. R&D ﬁrms hirelabor to engage in R&D, which improves products’ quality. Producing ﬁrms hire labor to produce products, gain
proﬁts by selling the products and distribute those proﬁts to households as dividends.
In the main analysis, we consider the following situation. Initially, there exists a sector with a continuumof products, the “old sector”, and subsequently another sector with a continuum of products, the “new sector”,
emerges. The two sectors diﬀer in four respects: eﬃciency, diﬃculty and cost of R&D, and market size. The
old sector, which is denoted sector 0, is so mature that the cost of R&D is low; however, the probability ofsuccessful R&D is also low because a large amount of R&D has already been accumulated. This assumption
follows Segerstrom’s notion that the most obvious ideas are discovered ﬁrst, making it harder to develop further
ideas. The new sector, which is denoted sector 1, is so new that the cost of R&D is high; however, probability ofsuccessful R&D is high because little R&D has yet been accumulated.
In each sector, there are continuous industries indexed by j ∈ [0, 1] producing diﬀerentiated goods. Eachindustry produces goods with diﬀerent qualities, which are classiﬁed into countable quality generation m = 0, 1, . . ..

5

The goods are perfectly substitutes for one another in each industry. Innovation in an industry results in a newgeneration of goods.
We designate as leaders (producing ﬁrms) (incumbents) those ﬁrms that use the latest R&D in each industry
and are able to produce the state-of-the-art quality product in their industry. At the time of birth of each sector, we
assume that the state-of-the-art quality product in each industry is m = 0 and that the size of quality increments
induced by one successful R&D event is constant in any industry of any sector, which is denoted as λ > 1. Therefore,
in industry j of sector i at time t, state-of-the-art quality is determined by the number of successful R&D events
that have occurred in industry j of sector i from the birth of sector i until time t.
We designate as R&D ﬁrms those ﬁrms that attempt R&D investment in order to win a leadership position
through a higher quality product. They may invest in any industry of any sector. If an R&D ﬁrm produces
successful R&D in industry j of sector i, it can produce a state-of-the-art quality product in industry j of sector i,and thus becomes the leader in industry j of sector i. When the state-of-the-art quality in industry j of sector i is m
– 1 generation, denoted as q i (j, m − 1), the successful R&D ﬁrm can produce an m generation product, the qualityof which is denoted by q i (j, m) = λq i (j, m − 1). If we normalize the initial quality to unity (i.e., q i (j, 0) = 1), thenan m generation of the state-of-the-art quality product in industry j of sector i is expressed as q i (j, m) = λm .

2.2Households

The economy is populated by a ﬁxed number of identical households. Household members live forever and are
endowed with one unit of labor, which is supplied inelastically. The number of members in each household grows
exponentially at an exogenous rate n > 0. Assuming that the initial population is unity, then the population attime t is L(t) = ent . Each household maximizes its discounted utility:∫∞exp(−(ρ − n)t) ln u(t)dt,

U=

(1)

0

where ρ > n is the subjective discount rate and ln u(t) is the instantaneous utility per person at time t, which is
given by[∫1ln u(t) = γ

0

ln
0

∑

]
m 0

λ d (j, m, t) dj + γ

m

ln
0

6

[

∫1
1

∑
m

]
m 1

λ d (j, m, t) dj,(2)

where γ i is the market share of sector i, di (j, m, t) is the quantity demanded of m generation product produced
in industry j of sector i at time t.5 The ﬁrst term on the right-hand side (RHS) is the instantaneous utility fromconsuming the products of sector 0; the integration of this term is the instantaneous utility from consuming the
products of industry j of sector 0, and the summation of this term is the instantaneous utility from consuming
m generation products of industry j of sector 0. The second term on the RHS is the instantaneous utility fromconsuming products of sector 1, and so on. Because γ i is the market share of sector i, γ 0 + γ 1 = 1 is always
satisﬁed. If γ 0 = 1, then this instantaneous utility function is identical to that of Segerstrom (1998).
From (2), the demand function for the product with the lowest quality-adjusted price in industry j of sector i
is given bydi (j, t) =γ i c(t),,pi (j, t) for i = 0, 1

(3)where pi (j, t) is the price of the product that has the lowest quality-adjusted price in industry j of sector i at timet, di (j, t) is the quantity demanded of the product that has the lowest quality-adjusted price in industry j of sector
i at time t, and c(t) is households’ expenditure at time t.6 Given (3), intertemporal utility maximization yieldsċ(t)= r(t) − ρ,
c(t)

(4)

where r(t) is the interest rate at time t.

2.3

Product markets

We assume the product market is monopolistically competitive. In each industry, ﬁrms compete in price. Labor isthe only input in production. We assume that one unit of labor is required to produce one unit of output regardlessof quality level, and that there is no ﬁxed cost. We normalize the wage rate as numeraire. Thus, every ﬁrm has aconstant marginal cost of production equal to unity.
In each industry, there is one leader and a number of followers. These followers were formerly able to produce
the state-of-the-art quality product but cannot now. Because the quality increment from one successful R&D
event is constant, leaders can produce a product of at least λ higher quality compared with the followers. Because
the marginal cost of production is unity, followers never charge a price less than unity. Leaders can therefore
5 As seen from (2), we assume that the intersector and interindustry elasticities of substitution are unity, and that the intra-industry
elasticity of substitution is inﬁnity.
6 Because intra-industry elasticity of substitution is inﬁnity, households demand only the product that has the lowest quality-adjusted
price in each industry

7

monopolize demand for their industry’s product by charging a price less than λ. Because interindustry elasticity
of substitution is unity, leaders have no incentive to lower the price to less than λ.7 Thus the price of the productis then determined byp(t) = λ.

(5)Because one unit of labor is required to produce one unit of output and there is no ﬁxed cost, leaders in sector iearn the following proﬁt ﬂow:

π i (t) = γ i c(t)L(t)(1 −1),
λ for i = 0, 1,(6)where π i (t) is the leaders’ proﬁts in sector i at time t.8

2.4

R&D races

Labor is the only input to R&D. There is free entry into each R&D race. Any R&D ﬁrm that hires H i (t)li (j, k, t)dt
units of labor in industry j of sector i at time interval [t,t+dt] is successful in discovering a state-of-the-art product/with instantaneous probability I i (j, k, t) = li (j, k, t)Ai X i (t)dt , where H i (t) is the cost of R&D, Ai is the eﬃciency
of R&D and X i (t) is the diﬃculty of R&D in industry i. Assuming that the returns to engaging in the R&D race are
independently distributed across ﬁrms, across industries and over time, the industry-wide instantaneous probabilityof successful R&D is given by

I i (t)dt =Li (t)Ai
dt, for i = 0, 1,
X i (t)(7)

where I i (t)dt is industry-wide instantaneous probability of successful R&D in sector i, and Li (t)dt is the amountof labor engaging in R&D in sector i. Following Segerstrom (1998), the diﬃculty of R&D in sector i increases as
R&D in sector i accumulates. Then we assumeẊ i (t)= µI i (t), for i = 0, 1,
X i (t)(8)

7 We assume that the consumer strictly prefers the higher-quality good when the quality-adjusted price is same.
8 Note that all industries’ proﬁts in the same sector are identical because both the quantity demanded in a sector and their prices

are identical.

8where 0 < µ < 1 is exogenously given. We deﬁne si as the time of origin in sector i. We also assume that
X i (si ) = X̄ i , that is, the initial diﬃculty of R&D is identical for all industries in the same sector. Then, the
)
(
∫t i
i
i
µIυ dν .
diﬃculty of R&D in sector i at time t is X (t) = X̄ exp
sii

Let v (t) denote the leaders’ stock prices in sector i at time t.9i iEach R&D ﬁrm attempts to maximizeil (j,k,t)v i (t) A X
dt − H i (t)li (j, k, t)dt . If v i (t) XAi (t) > H i (t) ; then, li = ∞ is proﬁt maximizing, which cannot be the
i (t)
i

i

equilibrium. If v i (t) XAi (t) < H i (t) , then li = 0 is proﬁt maximizing. If v i (t) XAi (t) = H i (t) , then li ∈ (0, +∞) is
proﬁt maximizing. To sum up,

i

i

i

i

(t)
v i (t) = X (t)HAiwhen I i (t) > 0 for(t)v i (t) < X (t)HAi

2.5

when I i (t) = 0

i = 0, 1(9)for i = 0, 1.

No arbitrage condition

Over time interval dt, shareholders receive an income gain π i (t)dt and a capital gain (loss) v̇ i (t)dt in each industry of
sector i. However, with instantaneous probability I i (t)dt, a higher-quality product is discovered, the existing leader
goes out of business and shareholders suﬀer a total capital loss v i (t). Eﬃcient markets require that the expected
i

i

i

i

(t)v (t)= r(t).
return to holding the stock of the leader is equal to the risk-free interest rate r(t).10 Thus π (t)+v̇ (t)−Iv i (t)From (6), we obtainv i (t) =γ i ( λ−1λ )c(t)L(t)i(t)r(t) + I i (t) − vv̇i (t), for i = 0, 1 .

(10)

In further analysis, let us deﬁne indexes of stock prices, V (t) and V̂ (t), as follows:

V (t) = v 0 (t) + v 1 (t) ∀I 0 (t), ∀I 1 (t) ,(11)where V (t) is deﬁned as the sum of stock prices of leaders across sectors where the leaders are producing goods.

V̂ (t) = v 0 (t)when I 0 (t) > 0, I 1 (t) = 0,

01V̂ (t) = v (t) + v (t) when

0

(12)

1

I (t) > 0, I (t) > 0

9 As seen below, we interpret v i (t) as the expected discounted proﬁt for winning an R&D race in sector i at time t. From (6), proﬁts
are identical for all industries in the same sector. Thus, the leader’s stock price is the same in all industries in the same sector.
10 Assuming that shareholders are risk neutral and can diversify risk completely.

9

where V̂ (t) is deﬁned as the sum of stock prices of leaders across sectors where the leaders are producing goods
and R&D ﬁrms are carrying on R&D.11

2.6

The labor market/
Because one unit of labor is required to produce one unit of output, γ i c(t)L(t) λ workers are employed by the/leader in sector i. X i (t)H i (t)I i (t) Ai workers are employed by R&D ﬁrms in sector i. Because households supply
one unit of labor inelastically, we obtain

X 0 (t)H 0 (t)I 0 (t) X 1 (t)H 1 (t)I 1 (t) c(t)L(t)= L(t).
+
+
A0
A1
λ

2.7

(13)

Growth rate of productivity

Because we assumed that the quality of 0 generation products for all industries of any sector is unity and that thesize of an increment of one successful R&D event is λ, the quality of the product manufactured by the leader ofindustry j of sector i is determined by the number of times that successful R&D occurs in that industry. We deﬁneQ(t) as the weighted sum of the quality of products across all industries of all sectors, and express Q(t) as follows:ln Q(t) =

1
∑
i=0

∫1
γ

i

i

ln λm (j,t) dj =1∑

γ i Φ(i, t) ln λ.

(14)

i=0

0

where mi (j, t) is the generation of state-of-the-art quality in industry j of sector i at time t, and Φ(i, t) =∫tsi

I(i, υ)dυ

[]is the expected value of successful R&D events that will occur in sector i in time interval si , t .
Deﬁne g(t) as the growth rate of the weighted sum of product qualities. From (14), we obtaing(t) ≡Q̇(t)= γ 0 I 0 (t) ln λ + γ 1 I 1 (t) ln λ.
Q(t)

(15)

Following Grossman and Helpman (1991), we interpret g(t) as the growth rate of productivity.12

11 We deﬁne two types of stock price index to deal with the problem of the timing of stock listing (the problem is when we should
evaluate new-sector incumbents’ stock prices.). Although the above deﬁnitions are not complete, they are suﬃcient for analyzing arough motion of stock price.
12 From (2) and (3), we obtain ln u(t) = ln Q(t) + γ 0 ln Lp0 (t) + γ 1 ln Lp1 (t), where Lpi is labor engaging in manufacturing in sector
i. Interpreting u(t) as the ﬁnal output, a higher Q(t) means that more output can be produced with the same input. Therefore, we
can take Q(t) as Total Factor Productivity (TFP )and g(t) as the growth rate of TFP.

10

3

Equilibrium

In this section, we derive the equilibrium path in two cases. In one case, R&D is done only in sector 0 (i.e., I 0 (t) > 0
and I 1 (t) = 0), and in the other case, R&D is done in both sectors (i.e., I 0 (t) > 0 and I 1 (t) > 0).

3.1

The case in which I 0 (t) > 0 and I 1 (t) = 0

First, derive the equilibrium path in the case where I 0 (t) > 0 and I 1 (t) = 0. Deﬁne the adjusted diﬃculty of R&D//
in sector i: y i (t) ≡ X i (t)H i (t) Ai L(t) Diﬀerentiating this equation with respect to t yields ẏ i (t) y i (t) = I i (t)µ − nfrom (8). Because I 1 (t) = 0, we obtain y 0 (t)I 0 (t) + c(t)λ = 1 from (13). Thenẏ 0 (t) =(λ − c(t))µ − y 0 (t)n.
λ(16)

γ 0 ( λ−1 )c(t)L(t)λBecause I 0 (t) > 0, we obtain v 0 (t) = y 0 (t)L(t) from (9). We also obtain v 0 (t) = r(t)+(1−µ)I
0 (t) from (8) and (10).

[ 0]1Combining these two equations and using y 0 (t)I 0 (t)+ c(t)λ = 1, we obtain r(t) = λy 0 (t) γ (λ − 1)c(t) − (1 − µ)(λ − c(t)) .
Substituting this equation into (4),ċ(t) =

1[(γ 0 λ + γ 1 − µ)(c(t))2 − (1 − µ)λc(t)] − ρc(t).
λy 0 (t)(17)

This economy’s dynamics are described by (16) and (17) in (y 0 , c) space, as depicted in ﬁgure 1. The verticalaxis represents consumption per capita, c, which is a jump variable, and the horizontal axis represents the adjusted
diﬃculty of R&D in sector 0, y 0 , which is a state variable. From (16), the ẏ 0 (t) = 0 curve is downward-sloping.
When the point (y 0 , c) is to the left (right) of the ẏ 0 (t) = 0 curve, then ẏ 0 (t) > 0 (ẏ 0 (t) < 0). From (17), theċ(t) = 0 curve is upward-sloping. When the point (y 0 , c) is to the left (right) of the ċ(t) = 0 curve, then ċ(t) > 0(ċ(t) < 0).
c(t)ċ(t) = 0ẏ 0 (t) = 0
y 0 (t)Figure 1. Phase diagram where I 0 (t) > 0 and I 1 (t) = 0.

We now derive the value of the balanced-growth equilibrium. Because the balanced-growth equilibrium is11

determined by the intersection of the ẏ 0 (t) = 0 curve and the ċ(t) = 0 curve, we obtainy 0∗ =

γ 0 µ(λ − 1),µρ − µn + n(γ 0 λ + 1 − γ 0 )

(18)

c∗ =

λ(µρ − µn + n),µρ − µn + n(γ 0 λ + 1 − γ 0 )

(19)

and

from (16) and (17), where the steady-state equilibrium level is denoted by “*” in the present case (where I 0 (t) >0 and I 1 (t) = 0).13 Because c∗ is constant in the balanced-growth equilibrium, we obtain r∗ = ρ from (4).
/Substituting (18) and (19) into (13) with I 1 (t) = 0 and y 0 (t) ≡ X 0 (t)H 0 (t) A0 L(t), we obtain

I 0∗ =

n
,
µ

(20)

and from (15), we obtainng ∗ = γ 0 (ln λ) .
µ(21)

µγ 0 (λ − 1)L(t),µρ − µn + n(γ 0 λ + 1 − γ 0 )(22)(µρ − µn + n)
µγ 1 (λ − 1)L(t).
(µρ − µn) µρ − µn + n(γ 0 λ + 1 − γ 0 )(23)From v 0 (t) = y 0 (t)L(t) and (18), we obtainv 0∗ (t) =and from (10) and r∗ = ρ, we obtainv 1∗ (t) =Substituting (22) and (23) into (11), we obtain

V ∗ (t) =(µρ − µn + nγ 1 )(λ − 1)L(t).
(ρ − n) [µρ − µn + n(γ 0 λ + 1 − γ 0 )]13 The equilibrium is saddle-path stable. Steger (2003) examines the stability conditions carefully.

12

(24)

From (12), we obtainV̂ ∗ (t) =3.2µγ 0 (λ − 1)L(t).
µρ − µn + n(γ 0 λ + 1 − γ 0 )(25)The case in which I 0 (t) > 0 and I 1 (t) > 0

In this subsection, we derive the equilibrium path in the case where I 0 (t) > 0 and I 1 (t) > 0. Because I 0 (t) > 0
γ 0 ( λ−1 )c(t)L(t)λand I 1 (t) > 0, we obtain v 0 (t) = y 0 (t)L(t) and v 1 (t) = y 1 (t)L(t) from (9). We also obtain v 0 (t) = r(t)+(1−µ)I0 (t)γ 1 ( λ−1 )c(t)L(t)γ 0 ( λ−1 )c(t)L(t)

0
0
0
1
λ
λ
and v 1 (t) = r(t)+(1−µ)I
1 (t) from (8) and (10). Substituting v (t) = y (t)L(t) into v (t) = r(t)+(1−µ)I 0 (t) , v (t) =γ 1 ( λ−1 )c(t)L(t)λy 1 (t)L(t) into v 1 (t) = r(t)+(1−µ)I
1 (t) , eliminating r(t) to combine these two equations, and using (13), we obtain]} (λ−1)c(t)γ 0 y 1 (t) − γ 1 y 0 (t) (1−µ)y
0 (t) + λ − c(t) ,[{]} (λ−1)c(t)1
I 1 (t) = λ(y0 (t)+y
γ 1 y 0 (t) − γ 0 y 1 (t) (1−µ)y
1 (t))
1 (t) + λ − c(t) .
1
I 0 (t) = λ(y0 (t)+y
1 (t))[{(26)/
Substituting these equations into ẏ i (t) y i (t) = I i (t)µ − n, we obtain[{]} (λ−1)c(t)γ 0 y 1 (t) − γ 1 y 0 (t) (1−µ)y
0 (t) + λ − c(t) − n,[{]} (λ−1)c(t)ẏ 1 (t)
µ
1 0
0 1
=
γ
y
(t)
−
γ
y
(t)
+
λ
−
c(t)
− n.
1
0
1
1
y (t)
λ(y (t)+y (t))(1−µ)y (t)ẏ 0 (t)µy 0 (t) = λ(y 0 (t)+y 1 (t))(27)

We deﬁne z(t) as the sum of adjusted diﬃculty of R&D across sectors where R&D is conducted:

z(t) = y 0 (t) + y 1 (t) whenz(t) = y 0 (t)when

I 0 (t) > 0, I 1 (t) > 0,I 0 (t) > 0, I 1 (t) = 0;(28)

then, ż(t) = ẏ 0 (t) + ẏ 1 (t) when R&D is conducted in both sectors.14 Using (27), we obtainż(t) =(λ − c(t))µ− z(t)n.
λ(29)

14 Note that ż(t) = ẏ 0 (t) when R&D is conducted only in sector 0. Therefore, we replace y 0 (t) with z(t), and ẏ 0 (t) with ż(t) in ﬁgure

1.

13

1
[(λ − µ)c(t) − (1 − µ)λ].
Substituting v i (t) = y i (t)L(t) and (26) into (10), and solving for r(t), we obtain r(t) = λz(t)Substituting this equation into (4), we obtainċ(t) =

1[(λ − µ)(c(t))2 − (1 − µ)λc(t)] − ρc(t).
λz(t)

(30)

This economy’s dynamics are described by (29) and (30) in (z, c) space, seen in the upper part of ﬁgure 2. The
vertical axis represents consumption per capita, c, which is a jump variable, and the horizontal axis represents thesum of adjusted diﬃculty of R&D across sectors, z, which is a state variable. The upper part of ﬁgure 2 is similar
to ﬁgure 1, but there are two diﬀerences between the ﬁgures. First, the horizontal axis of ﬁgure 1 represents y 0 (t),but that of ﬁgure 2 represents z(t).15 Second, the slope and the intercept of the ċ(t) = 0 curve of ﬁgure 1 aregreater than those of ﬁgure 2.16
From the upper part of ﬁgure 2, we can examine the macro variables, including consumption per capita andthe sum of stock prices. However, we are also interested in each sector’s variables, including the incumbents’stock prices in sector i and the industry-wide instantaneous probability of successful R&D in sector i. Deﬁne/
Y (t) ≡ y 1 (t) y 0 (t) as the ratio of adjusted diﬃculty of R&D in sector 1 to that of sector 0. From (27), we obtainẎ (t) =c(t)µ(λ − 1)γ 0
λ(1 − µ)y 0 (t)

{ 1
}
γ
−
Y
(t)
.
γ0

(31)

The dynamics of Y (t), which is a state variable, are illustrated in the lower part of ﬁgure 2. The vertical axis
represents the adjusted diﬃculty of R&D in sector 1, y 1 , and the horizontal axis represents the adjusted diﬃcultyof R&D in sector 0, y 0 . Because Y (t) is the ratio of y 1 (t) to y 0 (t), the slope of the line connecting the origin
1

and any point in (y 0 (t), y 1 (t)) space represents Y (t). From (31), the Ẏ (t) = 0 curve is Y (t) = γγ 0 . That is, theratio of adjusted diﬃculty of R&D in sector 1 to that of sector 0 is equal to the ratio of market share of sector
1

1 to that of sector 0. When Y (t) is smaller (greater) than γγ 0 , Ẏ (t) > 0 (Ẏ (t) < 0). That is, when the ratioof adjusted diﬃculty of R&D is smaller (greater) than the ratio of market share at time t, the ratio of adjusted
diﬃculty of R&D increases (decreases) at time t (i.e., industry-wide instantaneous probability of successful R&D
1

1

in sector 1 is greater (smaller) than that for sector 0). This is because when Y (t) < γγ 0 (Y (t) > γγ 0 ), sector 1 ismore (less) attractive than sector 0 because of the relatively high (low) probability of successful R&D or becauseof the relatively high (low) revenue from becoming the leader, and so more R&D ﬁrms are willing to conduct
investigations in sector 1 (0). This process continues until the ratio of adjusted diﬃculty of R&D is equal to the
15 However, y 0 (t) can be replaced with z(t) and ẏ 0 (t) with ż(t) in ﬁgure 1 from (28).
16 Note that λ > 1.

14

ratio of market share.
c(t)ċ(t) = 0(z ∗∗ , c∗∗ )(z0 , c0 )ż(t) = 0(y00 , y01 )0(y 0∗∗ , y 1∗∗ )z(t)y 0 (t)1

y + y = z0
Ẏ (t) = 0
y 0 + y 1 = z ∗∗y 1 (t)Figure 2. Phase diagram in the case where I 0 (t) > 0 and I 1 (t) > 0.

We give an example to promote better understanding of this dynamics. Suppose that the economy is initiallyat (z0 , c0 ) in (z,c) space and at (y00 , y01 ) in (y 0 , y 1 ) space of ﬁgure 2. Because z0 = y00 + y01 when I 0 (t) > 0 and
I 1 (t) > 0, (y00 , y01 ) is on the straight line that passes through the point (z0 , 0) and has a slope of forty-ﬁve degrees.
Because (z0 , c0 ) is to the left of the ż(t) = 0 curve and the ċ(t) = 0 curve, the economy moves northeast in (z,c)
space. Because (y00 , y01 ) is to the right of the Ẏ (t) = 0 curve, Y (t) increases. Thus, the economy moves to theẎ (t) = 0 curve satisfying y 0 (t) + y 1 (t) = z(t) in (y 0 , y 1 ) space, as depicted in ﬁgure 2. In the long run, becausez(t) converges to z ∗∗ , (y 0 (t), y 1 (t)) converges to (y 0∗∗ , y 1∗∗ ), which is the intersection of the Ẏ (t) = 0 curve andthe y 0 (t) + y 1 (t) = z ∗∗ curve. “**” denotes the steady-state equilibrium level in which I 0 (t) > 0 and I 1 (t) > 0.
Next we ﬁnd the value of the balanced-growth equilibrium.17 Because the balanced-growth equilibrium isdetermined by the intersection of the ż(t) = 0 curve and the ċ(t) = 0 curve, from (29) and (30) we obtainµ(λ − 1)µρ − µn + nλ(32)λ(µρ − µn + n).
µρ − µn + nλ(33)z ∗∗ =andc∗∗ =From (18) and (32), we conﬁrm that z ∗∗ > y 0∗ = z ∗ , and from (19) and (33), we conﬁrm that c∗∗ < c∗ .18
That is, the proportion of workers in R&D (manufacturing) is greater (smaller) when I 0 (t) > 0, I 1 (t) > 0 than17 The equilibrium is saddle-path stable.
18 Because λ > 1, γ 0 λ + 1 − γ 0 < λ.

15when I 0 (t) > 0, I 1 (t) = 0 from (13). Because c∗∗ is also constant in the balanced-growth equilibrium underI 0 (t) > 0, I 1 (t) > 0, we obtain r∗∗ = ρ from (4). From (31), we obtain

y 1∗∗
γ1
= 0.
0∗∗
y
γ

(34)

γ i µ(λ − 1)µρ − µn + λn for i = 0, 1.

(35)

Y ∗∗ =

Substituting (34) into (32), we obtainy i∗∗ =

From (18) and (35), we conﬁrm y0∗∗ ≤ y0∗ . Substituting (33) and (35) into (26), we obtain

I i∗∗ =nµ for i = 0, 1,(36)

and from (15), we obtainng ∗∗ = (ln λ) .
µ(37)From (21) and (37), we conﬁrm g ∗∗ > g ∗ .19 That is, the growth rate of productivity is greater when I 0 (t) >0, I 1 (t) > 0 than when I 0 (t) > 0, I 1 (t) = 0, because the proportion of workers engaging in R&D is greater whenI 0 (t) > 0, I 1 (t) > 0. Because v i (t) = y i (t)L(t), i = 0, 1 from (9), we obtainv i∗∗ (t) =γ i µ(λ − 1)L(t).
µρ − µn + λn(38)

From (22) and (38), we conﬁrm v 0∗∗ (t) < v 0∗ (t). From (23) and (38), we conﬁrm v 1∗∗ (t) < v 1∗ (t). That is, theequilibrium value of the incumbents’ stock price in both sectors is greater when I 0 (t) > 0, I 1 (t) = 0 than whenI 0 (t) > 0, I 1 (t) > 0. From (11), we obtainV ∗∗ (t) = V̂ ∗∗ (t) =µ(λ − 1)L(t).
µρ − µn + λn

(39)

From (24) and (39), we obtain V ∗∗ (t) < V ∗ (t). From (25) and (39), we obtain V̂ ∗ (t) < V̂ ∗∗ (t).

19 From (21) and (37), we conﬁrm that the growth rate of productivity is never inﬂuenced by the eﬃciency of R&D, Ai . This result
is diﬀerent from that in Boucekkine and de la Croix (2003), who state that the change in the eﬃciency of R&D aﬀects the long-run
growth rate.

16

4Comparative Dynamics

In this section, we analyze the transition dynamics caused by an exogenous drastic technological breakthrough,which leads to the emergence of the new sector. The process of the emergence of a new technology is classiﬁed
into three stages. Stage 1 is prior to the emergence of the new technology. In stage 1, there is only the old sector,
sector 0 (i.e., γ 0 = 1). In this stage, households can only consume products of sector 0 and both manufacturing
and R&D are carried out only in sector 0. When the economy is in stage 1, an unexpected major technologicalbreakthrough happens and a new sector, sector 1, emerges, which we deﬁne as the point at which the economy
enters stage 2. Stage 2 follows the emergence of a new technology, but the new technology is so new that R&D
ﬁrms are not yet willing to carry out R&D in the new sector. In stage 2, I 0 (t) > 0 and I 1 (t) = 0 (as discussed in
section 3.1). In this stage, households consume the products of both sectors 0 and 1 and manufacturing is carriedout in both sectors; however, R&D is conducted only in sector 0. When the economy is in stage 2, unexpectedmajor technological progress happens (cost of R&D falls drastically), which we deﬁne as the point at which theeconomy enters stage 3. By stage 3, the new technology have prevailed. In stage 3, I 0 (t) > 0 and I 1 (t) > 0 (asdiscussed in section 3.2). In this stage, households consume products of both sectors and manufacturing and R&D
are carried out in both sectors. This section analyzes each stage in more detail.

4.1

Stage 1: Before the emergence of new technology

Before the emergence of a new technology, the economy is in a balanced-growth equilibrium. 20 The economy ofstage 1 can be characterized by the present model with γ 0 = 1, which is identical with Segerstrom s (1998) model. Inµ(λ−1)
nn
0
0stage 1, the main endogenous variables are c(t) = λ(µρ−µn+n)µρ−µn+λn , y (t) = z(t) = µρ−µn+λn , I (t) = µ , g(t) = (ln λ) µ ,µ(λ−1)L(t)and V (t) = V̂ (t) = µρ−µn+λn, from (33), (35), (36), (37), and (39). This is point A in (z(t), c(t)) space and point

A’ in (y 0 (t), y 1 (t)) space of ﬁgure 3.21 Point A is the intersection of the ż(t) = 0 curve: (λ−c(t))µ− z(t)n = 0 and
λ1[(λ − µ)(c(t))2 − (1 − µ)λc(t)] − ρc(t) = 0.22 Point A’ is the intersection of the horizontal
the ċ(t) = 0 curve: λz(t)µ(λ−1)
axis and the z(t) = (µρ−µn+λn)line.

20 This assumption is based on the fact that major technological breakthroughs are rare.
21 y 1 (t) = 0 in stage 1 because sector 1 does not yet exist.
22 Note that when γ 0 = 1, the ż(t) = 0 curve is identical to the ẏ 0 (t) = 0 curve.

17ċ(t) = 0 (the cace of I 1 (t) = 0)

c(t)
B

D

C

ċ(t) = 0 (the cace of I 1 (t) > 0)AEFż(t) = 0E’z(t)y 0 (t)A’

F’ŷ 1 (t)Ẏ (t) = 0
C’y 1 (t)B’Figure 3. The movement of key variables in a phase diagram.
Figure 4 depicts the movements of the variables. The vertical axis represents the endogenous variables and thehorizontal axis represents time, t. The panel on the left depicts the movements of key variables, the middle panel
depicts the movements of variables related to the growth rate of productivity, and the panel on the right depicts
the movements of variables related to the stock price.

18

v 0 (t)/L(t)

c(t)

t
z(t)

t

y 0 (t)v 1 (t)/L(t)I 1 (t)tlnV(t)tttty 1 (t)g(t)lnV̂ (t)tstage1 stage2
τ2
τ1stage3t

I 0 (t)tstage1
τ1stage2
τ2stage3tstage1
τ1stage2stage3
τ2

Figure 4. Summary of the movement of key and economically important variables.

4.2

Stage 2: Before the start of R&D in the new sector

Stage 2 covers the period between the time at which a new technology emerges, τ1 , and the beginning of R&D in
the new sector. To guarantee the existence of this period, we assume that the cost of R&D in the new sector is
too high for R&D ﬁrms for several periods after the new technology emerges, τ2 .23 This assumption is based on
the fact that the costs of innovative activity in technologies such as electricity and computing were very high forsome time following the initial emergence of these new technologies, which discouraged people from using these
new technologies; it is also based on the idea that workers engaging in R&D do not know enough about the newtechnology to be able to improve the quality of products based on the new technology for that interval of timeafter the breakthrough.24 Here, this assumption is reﬂected in the high value of H 1 (t), which means a high valueof adjusted diﬃculty of R&D in sector 1, y 1 (t). ŷ i (t) is deﬁned as the critical value for the conduct of R&D sectori(t)from (9).2526 If y i (t) ≤ ŷ i (t), then R&D is conducted in sector i at time t; otherwise,i, for which ŷ i (t) = vL(t)R&D is not conducted in sector i. In stage 2, the inequality y 1 (t) > ŷ 1 (t) is always satisﬁed because of the highvalue of H 1 (t). Stage 2 is where I 0 (t) > 0 and I 1 (t) = 0, as discussed in section 3.1. In stage 2, the ċ(t) = 0 curveis expressed as (17), and the ż(t) = 0 curve is expressed as (16) with z(t) = y 0 (t).27 Both the intercept and the

23 If the cost of R&D is not too high, the economy goes directly from stage 1 to stage 3.
24 See Jovanovic and Rousseau (2005).
25 The critical value ŷ i (t) depends on time t because leaders’ proﬁts, interest rate and economy size vary over time.
26 If R&D is conducted in sector i, the adjusted diﬃculty of R&D in sector i is equal to the critical value because of the free entry
condition.
27 Note that I 1 (t) = 0 in stage 2 and (28).

19slope of the ċ(t) = 0 curve are greater in stage 2 than those in stage 1, whereas the ż(t) = 0 curve does not change
between stages.
Figure 3 allows us to analyze the transition dynamics. At time τ1 , the economy is at point A in (z,c) space
and at point A’ in (y 0 , y 1 ) space. As seen in the upper part of ﬁgure 3, at time τ1 , the ċ(t) = 0 curve shifts
counterclockwise, and then the intersection of the ċ(t) = 0 curve and the ż(t) = 0 curve shifts to the left. Because
point A is below the ċ(t) = 0 curve in stage 2, point A is no longer the equilibrium. To remain on the saddle path,c(τ1 ) has to shift upward, and then the economy should shift from point A to point B. Because point B is below
the ċ(t) = 0 curve and above the ż(t) = 0 curve in stage 2, c(t) and z(t) gradually fall to the new equilibrium,0ρλy (t)+(1−µ)λwhich is the intersection of c(t) = − λn. 28
µ z(t) + λ and c(t) =(γ 0 λ+γ 1 −µ)

We turn now to the lower part of ﬁgure 3. When the new technology emerges, y 1 takes a value such thaty 1 (τ1 ) > ŷ 1 (τ1 ). The economy then shifts from point A’ to point B’. Because y 0 (t) = z(t) throughout stage 2, y 0 (t)/decreases together with z(t). Because I 1 (t) = 0 in stage 2, ẏ 1 (t) y 1 (t) = −n. Then y 0 (t) and y 1 (t) decrease alongarrow B’C’.29 These stage 2 movements can be seen in the left-hand panel of ﬁgure 4.
Lemma 4.1. Summary of movement of key variables in stage 2• Consumption per capita, c, shifts upward at the emergence of the new technology, but decreases throughout
stage 2 without ever falling below its level in stage 1.
• The adjusted diﬃculty of R&D in sector 0, y 0 = z, remains unchanged at the emergence of the new technology,
but decreases throughout stage 2.
In the above analysis, we conﬁrm the movement of key variables in stage 2. Our next task is to analyze themovement of economically important variables such as the growth rate of productivity and stock price of old-sector
incumbents in stage 2.
Because the equilibrium value of c in stage 2 is greater than that in stage 1 and c(t) is decreasing throughout
stage 2, then consumption per capita in stage 2 is always strictly greater than that in stage 1. Therefore, theproportion of labor in R&D (manufacturing) in stage 2 is always strictly smaller (greater) than in stage 1.30
This result is related to the changes in the adjusted diﬃculty of R&D in sector 0, y 0 (t), in stage 2. Because y 0 (t)is decreasing and the size of the decrement of y 0 (t) decreases as it approaches equilibrium, I 0 (t) < nµ always holds in
28 Because ċ(t) < 0 and the size of the decrement of c(t) decreases as it approaches the stage 2 equilibrium, r(t) < ρ is always satisﬁed
and r(t) is increasing throughout stage 2 from (4).
11
1
29 In stage 2, ˆ˙y (t) = v̇ (t) − n. v̇ (t) < 0 is also satisﬁed, which is discussed later. Thus y 1 (t) is never less than ŷ 1 (t) in stage 2 asŷ 1 (t)v 1 (t)v 1 (t)long as the cost of R&D, H 1 (t), remains constant.
0
0
0
1
1
1
30 Dividing (13) by L(t), we obtain X (t)H (t)I (t) + X (t)H (t)I (t) + c(t) = 1. Thus, the proportion of labor in manufacturing andλA0 L(t)
A1 L(t)c(t)c(t)
R&D is λ and 1 − λ respectively.

20

stage 2 and I 0 (t) approaches I 0∗ = nµ from below throughout stage 2. Then, on the only feasible path, I 0 (t) shifts
downward at the emergence of the new technology and increases throughout stage 2. That is, the instantaneous
probability of successful R&D in sector 0 in stage 2 is always strictly less than in stage 1, but increases throughout
stage 2. We interpret this result intuitively as follows: at time τ1 , the proﬁts of leaders in sector 0 fall because sector
1 takes market share from sector 0, which means that the rewards for successful R&D in sector 0 decrease. The
number of R&D ﬁrms trying to enter sector 0 decreases, which means a decrease in the amount of labor engaged
in R&D in sector 0. The industry-wide instantaneous probability of successful R&D in sector 0 falls, which isa downshift of I 0 (t) at time τ1 . However a lower value of I 0 (t) leads to a decrease in the adjusted diﬃculty of
R&D in sector 0, which renders innovative activity more proﬁtable in sector 0.31 The number of R&D ﬁrms trying
to enter sector 0 increases throughout stage 2, which means that the industry-wide instantaneous probability ofsuccessful R&D in sector 0 increases throughout stage 2.32 On the other hand, R&D is not conducted in sector 1
in stage 2 because of the high cost of R&D. Thus, from (15), we obtain the important result that the growth rateof productivity is strictly less throughout stage 2 than in stage 1. This result is consistent with the stylized fact
known as “the productivity slowdown,” as depicted in the middle panel of ﬁgure 4.
In this model, the productivity slowdown occurs because of the following. The ﬁrst cause is the decrease in theindustry-wide instantaneous probability of successful R&D in sector 0. Sector 0 leaders’ proﬁts decrease because
sector 1 takes market share from sector 0, which decreases the incentive to become the leader in sector 0. Thisdecreases the amount of labor in R&D in sector 0, which in turn decreases the instantaneous probability of successful
R&D in sector 0.33 The second cause is the absence of R&D in sector 1, a consequence of the assumption that thecost of R&D is too high for several periods following the emergence of the new technology. From the evidence in
Jovanovic and Rousseau (2005), this assumption seems plausible. However, it may be that the initial cost of R&D
in the new sector is not high enough relative to its market share, which means that the economy goes directly fromstage 1 to stage 3, with no stage 2. This possibility suggests that whether a technological breakthrough causes a
productivity slowdown depends on the initial cost of R&D and the market share of the new sector.
Because y 0 (t) decreases over stage 2, v 0 (t) also decreases over stage 2 from (9). v 1 (t) also decreases over stage
2.34 Thus, both V (t) and V̂ (t) decrease in stage 2 from (11) and (12). Because y 0 (t) is not changed by the
31 Although the accumulation of R&D in sector 0 continues while I 0 (t) > 0, the population grows faster than R&D accumulates instage 2. Thus, the adjusted diﬃculty of R&D in sector 0 decreases.
32 Note that because y 0 (t) decreases throughout stage 2, I 0 (t) never exceeds n .
µ
33 From (7) and (10), we see that I 0 (t) must decrease in order to satisfy (7) with equality.
1
34 From (10) and I 1 (t) = 0, we obtain v̇ (t)v 1 (t)c(t)= r(t) − γ 1 ( λ−1) v1 (t) L(t). This equation shows that v̇ 1 (t) increases with v 1 (t), r(t)λv̇ 1 (t)and decreases with c(t). In stage 2, r(t) increases and c(t) decreases. Thus, v1 (t) is increases throughout stage 2 as long as v 1 (t) doesv̇ 1 (t)v̇ 1 (t)not decrease. Then, if v1 (t) > 0, v 1 (t) goes to inﬁnity and never converges to the equilibrium. Therefore, v1 (t) < 0 must always holdin stage 2 if convergence is to occur. Therefore, the only feasible path of v 1 (t) is that v 1 (t) shifts upward at time τ1 and decreases

21emergence of a new technology, neither v 0 (t) nor V̂ (t) is changed by that event. Assuming that sector 1 leaders’
stocks are listed at the moment of the technology’s emergence, v 1 (t) is evaluated at time τ1 . Then V (t) shifts
upward to the same degree as v 1 (τ1 ) at time τ1 . These movements in stage 2 are depicted in the right-hand panelof ﬁgure 4.
What happens to v 0 (t) in stage 2? At time τ1 , the new sector emerges, and the old sector loses market share to
the new sector, which causes π 0 (t) to decline. From (10), v i (t) is an increasing function of π(t)i and a decreasing
function of I i (t). Thus the stock price of incumbents in the old sector decreases unless the replacement risk, I 0 (t),changes. However, R&D activity, I 0 (t), also decreases to satisfy the free entry condition, then v 0 (t) does not shiftdownward at τ1 .35 As seen above, I 0 (t) < nµ always holds in stage 2, in which case ẏ(t)0 < 0 also holds.36 Thismeans that the adjusted diﬃculty of R&D in sector 0 decreases throughout stage 2, which leads to more R&D
activity. Then I 0 (t) increases throughout stage 2. Because π 0 (t) remains small in stage 2, v 0 (t) decreases becauseof an increase in the replacement risk, I 0 (t). We note also that because c(t) decreases and r(t) increases throughout
stage 2, the decline in v 0 (t) and v 1 (t) will be exacerbated.37 As shown in the last paragraph, both V (t) and V̂ (t)decrease throughout stage 2, which is consistent with the empirical data. This shows that the aggregate stock pricedecreases regardless of the timing of new ﬁrms’ stock listing. Thus the decline of v 0 (t), which is caused by the
decline of the market share of sector 0, plays an important role in the decline of the aggregate stock price.
Proposition 4.1. Summary of the movements of economically important variables in stage 2, in which R&D is
not conducted in sector 1 after the emergence of a new technology.
• The growth rate of productivity, g, shifts downward at the moment of the emergence of the new technology,
and increases throughout stage 2; however, it never goes over the level of stage 1.
• The stock price of incumbents in the old sector, v 0 , decreases throughout stage 2.
• The sum of the stock price indexes, V and V̂ , decrease throughout stage 2.

4.3

Stage 3: After R&D in the new sector begins

Third, we analyze stage 3, which is the stage after R&D begins in the new sector. We refer to the moment when
R&D ﬁrms in new sector begin to carry out R&D as τ2 . When the economy is in stage 2, unexpected drastic
technological progress occurs (cost of R&D falls drastically), which forces the economy to move from stage 2 tothroughout stage 2.
35 From (9), v i (t) is a function of the state variable, X(t)i , and exogenous variables, H i (t) and Ai , when I i (t) > 0. Thus v i (t) does
not shift unless the exogenous variables change.
/
36 Note that a diﬀerential equation of y(t)i is ẏ i (t) y i (t) = I i (t)µ − n
37 From equation (10), v i (t) is an increasing function of c(t) and a decreasing function of r(t).

22stage 3. In our model, this phenomenon reﬂects an unexpected suﬃcient drop in H 1 (t), which causes an unexpected//
drop in y 1 (t) satisfying y 1 (t) ≤ ŷ 1 (t) and y 1 (t) y 0 (t) < γ 1 γ 0 . Although this assumption seems to be restrictive,
we obtain similar results by extending this model to the model where the cost of R&D decreases due to learningby doing.38 However, as this expansion is somewhat complex and the results do not change dramatically, for now
we regard the cost of R&D as an exogenous variable.
Because of the unexpected exogenous shock, y 1 (t) does not become greater than ŷ 1 (t); hence, we express the
economy of stage 3 as the case where I 0 (t) > 0 and I 1 (t) > 0, which is discussed in section 3.2. Then, in stage 3,
the ċ(t) = 0 curve is expressed as (30), and the ż(t) = 0 curve is expressed as (29). Both the intercept and theslope of ċ(t) = 0 curve are smaller in stage 3 than those in stage 2, whereas the ż(t) = 0 curve is identical in bothstages. On the other hand, the ċ(t) = 0 curve and the ż(t) = 0 curve of stage 3 and those of stage 1 are completely
identical.
We then analyze the economic movement by using ﬁgure 3 again. At time τ2 , we consider that the economy isat point C in (z,c) space and at point C’ in (y 0 , y 1 ) space (i.e., the unexpected shock occurs when economy is at
point C and point C’). First, we examine the upper part of ﬁgure 3. At time τ2 , ċ(t) = 0 curve shifts clockwise, andthe intersection of the ċ(t) = 0 curve and the ż(t) = 0 curve shifts to the right. Because I 0 (t) > 0 and I 1 (t) > 0
by the drastic technological progress, z(t) should be evaluated as z(t) = y 0 (t) + y 1 (t) (c.f., z(t) is evaluated asz(t) = y 0 (t) in stage 2). Then z(t) shifts to the right to the same degree as y 1 (τ2 ), and the economy shifts frompoint C to point D. Because point D is above the ċ(t) = 0 curve and above the intersection of the ċ(t) = 0 curve
and the ż(t) = 0 curve of stage 3, then point D is no longer on the saddle path. To return to the saddle path, c(τ2 )has to shift downward, and then the economy should shift from point D to point E. Because point E is above theċ(t) = 0 curve and below the ż(t) = 0 curve, c(t) and z(t) increase gradually to the new equilibrium point F, which
ρλz(t)+(1−µ)λ 39.
is the intersection of c(t) = − λnµ z(t) + λ and c(t) =λ−µWe next examine the lower part of ﬁgure 3. At the moment that the unexpected exogenous shock occurs, the//adjusted diﬃculty of sector 1 drops dramatically, which satisﬁes y 1 (τ2 ) ≤ ŷ 1 (τ2 ) and y 1 (τ2 ) y 0 (τ2 ) < γ 1 γ 0 . The
economy then shifts from point C’ to point E’. In stage 3, the dynamics of y 0 and y 1 are determined by (31).
//Because we assumed that y 1 (t) y 0 (t) < γ 1 γ 0 , the ratio of adjusted diﬃculty of R&D of sector 1 to that of sector
0, Y (t), increases in stage 3. Note that from (18) and (35), y 0∗∗ < y 0∗ and y 0 is a state variable; therefore, y 0
1

Ḣ 1 (t) = −ν 1 XA1 (t) Lp1

when

H 1 (t) > H̄(t), where H̄(t) and ν 1 are parameterswhen H 1 (t) = H̄(t)and Lp1 is the amount of labor engaging in manufacturing in sector 1. Then, the cost of R&D, H 1 (t), becomes an endogenous variable
and the regime switch from stage 2 to stage 3 is determined endogenously.
39 Because ċ(t) > 0 and the size of increment of c(t) decreases when approaching the equilibrium in stage 3, r(t) > ρ is always satisﬁed
and r(t) decreases throughout stage 3 from (4).
38 We deﬁne the diﬀerential equation of H 1 (t) asḢ 1 (t) = 0

23

decreases in stage 3. Because z(t) increases and y 0 (t) decreases in stage 3, y 1 (t) must increase in stage 3. Theny 0 (t) and y 1 (t) move to F’, which is on the Ẏ (t) = 0 curve satisfying y 0∗∗ + y 1∗∗ = z ∗∗ . These movements are
depicted in stage 3 of the left-hand panel of ﬁgure 4.
Lemma 4.2. Summary of movement of key variables in stage 3• Consumption per capita, c, shifts downward the moment R&D in the new sector begins, but increases throughout stage 3. However, it is never above the level in stage 1.
• The sum of the adjusted diﬃculty of R&D across sectors, z, shifts upward the moment R&D in the new sector
begins, and increases throughout stage 3. However, it is never above the level in stage 1.
• The adjusted diﬃculty of R&D in sector 0, y 0 , remains unchanged the moment R&D in the new sector begins,but decreases throughout stage 3.
• The adjusted diﬃculty of R&D in sector 1, y 1 , shifts downward the moment R&D in the new sector begins,but increases throughout stage 3.
In the above analysis, we conﬁrm the movement of key variables in stage 3. Next, we analyze the movement of
economically important variables such as the growth rate of productivity, the value of old-sector leaders’ stock andthe sum of stock prices in stage 3.
Because the equilibrium value of consumption per capita in stage 3 is smaller than that in stage 2 and c(t)increases throughout stage 3, consumption per capita in stage 3 is always strictly smaller than that in stage 2.
Therefore, the share of labor engaging in R&D (manufacturing) in stage 3 is always strictly greater (smaller)
than in stage 2 from (13). Although the total share of labor engaging in R&D is greater than that in stage 2,
because ẏ 0 (t) < 0, the industry-wide instantaneous probability of successful R&D in sector 0 is always less thanthe equilibrium value, I 0∗∗ = n/µ, but increases throughout stage 3 to I 0∗∗ = n/µ. Although the impact of I 0 (t)at time τ2 is ambiguous, if stage 2 is long enough (i.e., I 0∗ ≃ n/µ ), then I 0 (τ2 ) shifts downward at the momentthat R&D begins in sector 1. On the other hand, because ẏ 1 (t) > 0 and the size of the increment to y 1 (t) > 0
decreases throughout stage 3, the industry-wide instantaneous probability of successful R&D in sector 1 is always
greater than the equilibrium value, I 1∗∗ = n/µ, and decreases throughout stage 3 to I 1∗∗ = n/µ. Then the only
feasible path, I 1 (t), shifts upward precisely at τ2 and decreases throughout stage 3. Therefore, I 0 (t) ≤ I 1 (t) isalways satisﬁed throughout stage 3, which is consistent with Ẏ (t) > 0 throughout stage 3. Because ż(t) > 0,Ẏ (t) > 0 and I 0 (t) < I 1 (t) throughout stage 3, g(t) = γ 0 I 0 (t) + γ 1 I 1 (t) > n/µ throughout stage 3 as depicted in24the middle panel of ﬁgure 4 (see Appendix). This result is consistent with the empirical fact that the growth rateof productivity returns to a high level after a lapse of some years from the emergence of new technology.
This result has a simple explanation. When the economy arrives at stage 3, the cost of R&D in sector 1 falls
dramatically. R&D ﬁrms in the new sector can then carry out R&D with a high probability of success (i.e., alow value of X 1 (t)) and with a low cost of R&D, (i.e. a low value of H 1 (t)), which induces a large volume of
R&D activity by R&D ﬁrms in sector 1. Thus, the number of workers engaging in R&D activity in the new sector
increases, which leads to a high industry-wide instantaneous probability of successful R&D in the new sector. By
contrast, the number of workers engaging in R&D in sector 0 decreases because R&D ﬁrms in sector 1 take workers
from R&D ﬁrms in sector 0, which reduces the industry-wide instantaneous probability of successful R&D in sector
0. However, the analysis in our appendix shows that the former eﬀect dominates latter, and thus the growth rateof productivity is high throughout stage 3.
Because y 0 (t) (y 1 (t)) decreases (increases) in stage 3, v 0 (t) (v 1 (t)) also decreases (increases) in stage 3 from
(7). Furthermore, because z(t) increases in stage 3, the stock price indexes, V (t) and V̂ (t), also increase in stage
3 from (11), (12) and (28). Because y 0 (t) does not change at τ2 , v 0 (t) does not change at that point either. v 1 (t)in stage 2 is always higher than in stage 3 because new-sector leaders face no threat of replacement. In stage 3,
these leaders face replacement because R&D commences in sector 1. Thus, the stock price of sector 1 leaders will
fall as soon as R&D begins in sector 1. Therefore, V (t) shifts downward at time τ2 . Consider instead the indexV̂ (t); because sector 1 leaders’ stocks are listed at R&D in sector 1 begins (i.e., v 1 (t) is evaluated at time τ2 ), thenV̂ (t) shifts upward to the same degree as v 1 (τ2 ) at time τ2 . These movements are shown in the right-hand panelof ﬁgure 4.
The reason v 1 (t) remains small and never returns to its stage 1 level is very simple: it is that π 1 (t) also remains
small and never returns to its stage 1 level because of the emergence of the new sector. Then the cause of theincrease in V (t) and V̂ (t) throughout stage 3 is the increase in v 1 (t) in stage 3. This comes mainly from thedecrease in I 1 (t) throughout stage 3. As the new sector matures, that is, as product quality in the new sector
improves, subsequent new-sector R&D becomes more diﬃcult. This reduces the probability of successful R&D in
the new sector (decreases the risk of new-sector leaders being replaced), which increases v 1 (t) throughout stage 3.
In addition, the value of the increment of v 1 (t) relies in part on the results that c(t) increases and r(t) decreases
throughout stage 3. Because the eﬀect of this increment dominates the eﬀect of the decline in v 0 (t), V (t) and V̂ (t)increase throughout stage 3.
Proposition 4.2. Summary of movements of economically important variables in stage 3, where R&D is conducted25

in sector 1• The growth rate of productivity, g, shifts upward when R&D in sector 1 begins, and decreases throughout stage
3. However, it never falls below the level in stage 1.
• The stock price of old-sector incumbents, v 1 , remains small and never returns to the level in stage 1.
• The sum of the stock prices, V and V̂ , increases throughout stage 3.

5Conclusion

Using a typical R&D growth model, we examined the transition dynamics of a technological breakthrough. We
showed that a technological breakthrough causes a productivity slowdown and a fall in the stock prices of existingﬁrms. Both these results are consistent with events of the 1970s. Our model provides an explanation for theseeﬀects of technological breakthroughs and shows that productivity slowdown occurs if and only if the initial costof R&D activity is high enough.
In this model, market share is exogenous and constant throughout the transition. However, the market shareof a new sector can be small initially and grows throughout the transition. This problem is caused by our utility
function having a unit elasticity of substitution between sectors. A utility function having any constant elasticity
of substitution between sectors will solve this problem. This is an important direction for future work.

6

Appendix

6.1

A1

We show that the growth rate of productivity in stage 3 is greater than the equilibrium value. Because ż(t) > 0 in/stage 3, and note that ẏ i (t) y i (t) = I i (t)µ − n, the following holds:ż(t) > 0
⇔ I 0 (t)y 0 (t)µ − ny 0 (t) + I 1 (t)y 1 (t)µ − ny 1 (t) > 0
1

0

y (t)
y (t)
n
1
⇔ I 0 (t) y0 (t)+y
1 (t) + I (t) y 0 (t)+y 1 (t) > µY (t)n⇔ I 0 (t) 1+Y1 (t) + I 1 (t) 1+Y(t) > µ .

26

(40)

Y (t)
Because I 0 (t) < I 1 (t), then I 0 (t) 1+Y1 (t) + I 1 (t) 1+Y(t) is an increasing function of Y (t). From Ẏ (t) > 0, we
1

obtain Y (t) < γγ 0 . Then the following holds:

Y (t)I 0 (t) 1+Y1 (t) + I 1 (t) 1+Y(t)< I 0 (t)

1
γ1
1+ γ
0

+ I 1 (t)

γ1
γ0
1
1+ γ
γ0
γ1
γ0
0
γ +γ 1γ0

1
1= I 0 (t) γ 0 +γ
1 + I (t)γ0

= I 0 (t)γ 0 + I 1 (t)γ 1 .

From (40) and (41), we obtain g(t) = γ 0 I 0 (t) + γ 1 I 1 (t) > n/µ

27

(41)

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