Boom and Bust Cycle of the Stock Market, and Economic Growth
in a Vintage Capital Model $Takeshi Kobayashi*

andMasaya Sakuragawa **We construct a growth model of overlapping generations with vintage capital. There exists anequilibrium that converges to the balanced growth path through endogenous fluctuations ofinvestment, consumption, and output in terms of the growth rate. When the technologicalchange arrives and a rise in productivity is embodied only in newly invested capital, theeconomy converges to a new balanced growth path with a higher growth rate of output, butwhen we interpret the price of existing old capital as the stock market capitalization, the rise inproductivity is accompanied by an initial decline in the stock market. Oscillatory equilibria aresupported as perfect-foresight equilibria in the present framework with finitely lived agents andcapital. Any oscillatory equilibrium is associated with the regime switch from an economy withboth young and old capital in use into the one with only old capital in use.

Keyword: Economic Growth, Vintage Capital, Technological Change.
JEL Classification: E13, E32, O33, O47.

$ We are greatly thankful to Kaoru Hosono, Hidehiko Ishihara, Boyan Jovanovic, Yukie Sakuragawa,Tadashi Shigoka, Noriyuki Yanagawa, and especially an anonymous referee for valuable comments.
* Department of Economics, Chukyo University, Nagoya, JAPAN, takeshi@mecl.chyukyo-u.ac.jp** Department of Economics, Keio University, Tokyo, JAPAN. masaya@econ.keio-cu.ac.jp.

1. Introduction
We construct a growing version of Diamond’s (1965) overlapping generations model withvintage capital. The idea of vintage capital is motivated by the insight on the recent
technological progress, called the “IT revolution,” that is supposed to have started with theinvention of the microprocessor around 1973. The new technology did not diffuse equally
among all firms, but favored newly established firms over incumbent ones [e.g. Hobjin and
Jovanovic (2001)]. We think that the vintage capital model can describe well a situation inwhich a rise in productivity is embodied only in newly invested capital, not in existing oldcapital.1
The main departure from the Diamond model is that capital does not depreciate
exponentially and is periodically replaced.2 In particular, we assume that capital depreciates
fully after two periods. We consider this assumption plausible. Setting the annual depreciationof capital to be 10%, capital survives by about 12.2% after twenty years, but only about 1.5%
after forty years.
At any instant of time, at most two types of capital, “young” capital and “old” capital, are inuse. There exists an equilibrium that converges to the balanced-growth path through endogenous
fluctuations of investment, consumption, and output in terms of the growth rate. A greater
investment in new capital of the current period implies a greater stock of old capital purchasedby agents of the next generation at the end of the subsequent period, which in turn tends tocrowd out new capital investment by the next generation. The counter-cyclical behavior
between new capital investments of the two successive periods is the driving force ofendogenous fluctuations of economic growth.3
1

A number of vintage-capital models describe the process of “creative destruction” associated withtechnological progress, including those of Caballero and Hammour (1994)(1996), Boldrin andLevine (2001), Greenwood et al (1997), Hobjin and Jovanovic (2001), and Laitner and Stolyarov(2003).
2
Although we focus on vintage of physical capital in this paper, we observe other types of vintage
capital models. Chari and Hopenhayn (1991) develop a model allowing for complementarity
between experienced and inexperienced workers in order to explain the empirically observed gradual
diffusion of technology. Fershtman, Murphy, and Weiss (1996), and Kremer and Thomson (1998)
construct growth models allowing for complementarity between different vintages of human capital.
3
Many vintage capital models derive the cyclical behavior of economic activities, including
Benhabib and Rustichini (1991), Caballero and Hammour (1994), Boucekkine, et al (1999), andothers. Benhabib and Rustichini (1991) study how nonexponential depreciation of capital can lead to1

We investigate the effects of technological change on the price of old capital and economic
growth. As we often observe in the information technology (IT) industries, we consider asituation in which technological progress is embodied only in new capital investment. The
unequal diffusion of new technology leads not only to improvement of the productivity of newcapital but also to the acceleration of endogenous obsolescence of old capital. The stock market
capitalization, defined by market values of old capital relative to GNP, initially declines. The
perverse behavior of the stock market disappears in the long run, and the economy finallyattains faster growth with a higher level of stock market capitalization. The technologicalchange accelerates economic growth accompanied by faster depreciation of existing capital.
The impact of the technological revolution on welfare is interesting. When there is anunanticipated technological revolution, young agents are strictly better off, while old agents arestrictly worse off if government intervention is not permitted. An appropriate transfer of income
through government intervention enables the technological revolution to attain Pareto
improvement allocation.
In the latter part, we explore the possibility that oscillatory equilibria are supported as
perfect-foresight equilibria in the present framework with finitely lived agents and capital. Any
oscillatory equilibrium involves the regime switch from an economy with both young and oldcapital in use into the one with only old capital in use. The regime switch involves the boom inthe price of old capital, accompanied by zero investment in new capital.4 Although the price ofold capital exhibits bubble-like behavior, the asset value reflects only economic fundamentals,
but does not include the “rational bubbles” defined by Tirole (1985).5
We use a model of finitely lived agents although many vintage models are constructed in a
setup with infinitely lived agents. One advantage of our model is to simplify the analysis in
showing the behavior of the stock market and its interaction with economic fundamentals whencycles in investment, as capital is periodically replaced. Boucekkine et.al (1997) (1998) characterize
the dynamics of capital replacement and economic fluctuations under endogenous scrapping.
4
Another perspective in the literature explains boom and crash in the stock market based oninvestors psychology or irrationality; see Bulow and Klemperer (1994), Lee (1998), and Zeira(1999).
5
Tirole (1985) develops a theoretical analysis linking economic growth with stationary bubbles thatnever explode in the model with finitely lived agents. See also Grossman and Yanagawa (1993) andCozzi (1998).
2the technological revolution arrives.
We finally comment on the related literature. We differ from Hobjin and Jovanovic (2001)
in two respects. First, we derive the endogenous obsolescence of old capital even by assumingthat the productivity of old capital remains unchanged, in contrast to Hobjin and Jovanovic(2001) who assume that old capital becomes less productive when technological progress occurs.
Second, we analyze the interaction among the stock market, capital accumulation, and growth.
Laitner and Stolyarov (2003), while holding views close to ours, apply the idea of old capitalobsolescence to the measurement problem of capital, succeeding in reconciling a fall in Tobin’s
Q with unchanging aggregate investment that arose during the period 1974-1984. Unlike theirs(2003), we develop an overlapping generations model, which seems to be more suitable forinvestigating the effects of an event that rarely occurs, such as revolutionary technologicalprogress. In addition, the model allows us to study the effects of the revolution on the welfare ofdifferent generations.
The paper is organized as follows. Section 2 describes the model. Section 3 analyzes theequilibrium of the balanced-growth path. Section 4 investigates the effect of the technologicalchange on the economy. Section 5 analyzes oscillatory equilibria when the boom in old capitalarises. Section 6 provides concluding remarks.

2. The Model
We consider an economy consisting of an infinite sequence of two-period-lived, overlapping
generations, plus an initial old generation. We let t=0,1,... index time. At each period acontinuum of new generation appears with unit mass. Young agents are identical and endowedwith one unit of labor, which is supplied inelastically in the labor market. The aggregate supplyof labor at each period is unity. Old agents are retired. The initial old agents are each endowedwith I −2 (>0) units of capital that was invested in period –2 and I −1 (>0) units of capital thatwas invested in period –1. No agents other than the initial old ones have any endowment of
capital or the final good at any period.
The preference of agents born at t (≥ 0) is given by3

log Cty + β log Cto+1 ,(1)where Ct (Ct +1 ) denote consumption when they are young (old), and β ( 0 < β ≤ 1) is theyosubjective discount factor. In the second period of life, each agent operates its own firm in orderto produce the final good according to the technology
1−γ

Yt = AK tγ L1t−γ k t , with 0 < γ < 1(2)where Yt is the final output, K t and Lt are capital and labor employed by the firm, k t is
the average capital-labor ratio across firms which represents technological spillovers on the
productivity of each firm, and A is the total factor productivity that is assumed constant overtime.
Agents have two means of saving the first-period income. The first is access to a
constant-returns-to-scale technology that converts xt units of the final good into at +1 xt
efficiency units of capital between periods, where at +1 represents a technological parameterthat is specific to the period-t investment. This assumption is meant to envisage a situation inwhich the technological change is embodied in new capital investment. The second is thepurchase of existing capital in the secondary market from agents of the preceding generation.
Capital depreciates fully after two periods.6 It turns out that, at the beginning of period t+1,
two types of capital may exist that differ in vintage, the capital good originally produced atperiod t, referred to as “young capital,” and the one produced at t-1, referred to as “old capital.”
After young capital is used as input for the final-good production, a portion δ (0 ≤ δ < 1) ofthat capital depreciates physically, and the remaining 1 − δ survives and is sold in thesecondary market as old capital.7 Young and old capital are perfect substitutes as inputs in the
production function of the final good.
The labor market and the capital market are assumed to be perfectly competitive. Since all
6

The following fact might justify this assumption. Setting the annual depreciation of capital at 10%,
capital survives by about 12.2% after twenty years, and only about 1.5% after forty years
7
The depreciation rate δ represents physical depreciation, and as will become clear below, should
be distinguished from economic depreciation that arises from devaluations of capital.
4

the firms are homogeneous, each of them chooses the same capital-labor ratio as the other. Since
the total labor force is unity, each young agent earns the amount equal to the wage rate, denotedby Wt , to satisfy

Wt = (1 − γ ) AK t ,(3)

where k t = K t Lt is used.
If agents invest one unit of the final good in the capital-production technology at period t,they produce at +1 efficiency units of capital at period t+1, receive at +1γA as rents, and sell
the remaining capital to agents of the subsequent generation. Letting pt +1 denote the price ofold capital measured in terms of physical unit, the gross rate of return from investing in young
capital, denoted by Rtnew+1 , isRtnew+1 = at +1γA + (1 − δ ) pt +1 .

(4)

Agents may alternatively purchase old capital in the secondary market. They purchase oldcapital at pt per physical unit, receive rent at γA in the next period and then liquidate it. Old
capital is no longer of use after the second-round production. The gross rate of return from oldcapital, denoted by Rtold+1 , is

Rtold
+1 =

at γA
.
pt

(5)

If young and old capital are both held, agents should be indifferent in their regard. The
oldno-arbitrage condition requires Rtnew+1 = Rt +1 , or equivalently

at +1γA + (1 − δ ) pt +1 =

at γA
.
pt

(6)

3. Equilibrium with Balanced-Growth Path
We conduct a general equilibrium analysis of this economy. In this section we investigate thedynamic behavior of an economy in which new and old capital are both held. In fact, theeconomy can switch to an economy in which only one kind of capital will be in operation, but
the analysis is left to Section 5.

5

From the preference given by (1), it is obvious that the aggregate savings, denoted by S t , aregiven byS t = sWt ,(7)

where s ≡ β (1 + β ) is interpreted to be the saving rate.
Let I t denote the aggregate amount of the final good that is channeled to the production of
new capital at period t.

The market clearing in the good market requires that the aggregate

savings equal the sum of the total investment in young capital, I t , and the aggregate market
value of the remaining old capital, (1 − δ ) pt I t −1 . It follows from (3) and (7) thats (1 − γ ) AK t = I t + (1 − δ ) pt I t −1 .

(8)

The aggregate capital at the beginning of period t+1 is composed of young capital invested atperiod t and old capital initially invested at period t-1 and traded in the secondary market at t,
given by

K t +1 = at +1 I t + (1 − δ )at I t −1 .

(9)pt ≡ pt a t denote theLet ηt ≡ I t I t −1 denote the gross growth rate of young capital and let ~price of old capital measured in terms of efficiency unit, which we may call a “stock price.”
Substituting (9) into (8) yields

η t = s (1 − γ ) A{at + (1 − δ )at −1

η t −1} − (1 − δ )at ~
pt .

(10)

By dividing both sides of (6) by at +1 , we rewrite the no-arbitrage condition (6) by

γA + (1 − δ ) ~pt +1 =γA.
at +1 ~
pt

(11)

Each side of (11) measures the rate of return from capital in terms of efficiency units.
Here we define perfect-foresight equilibria by assuming that the technological parameterat is constant over time, such that a ≡ at = at +1 = ... .

Definition 1
Assume that the technological parameter at is constant over time. Any perfect-foresightequilibrium of an economy where both young and old capital are held is a sequence of variables{ηt , ~pt }t∞=0 which satisfies (10), (11), and ~pt ≥ 0 , given η −1 > 0 .

6

First of all, we conduct a steady state analysis. The steady-state pair {η , ~p} is characterized by
the balanced growth path where investment, capital, and output grow at the rate η , while theprice of old capital is constant over time, satisfying

γA
γA + (1 − δ ) ~p = ~ ,ap
η = s(1 − γ )aA(1 +

and

1−δ

η

(12)

) − (1 − δ )a~
p.

(13)

We now analyze dynamic equilibria and obtain the following.

Proposition 1
A sufficient condition under which there exists a perfect-foresight equilibrium that converges tothe balanced-growth path is− γA + (γA) 2 + 4 (1 − δ )γA a s (1 − γ ) A~<p=.
2(1 − δ )
1−δ

(14)

Proof: In order to make the local analysis, we linearize the system (10) and (11) in theneighborhood of the steady state. Denoting the R.H.S. of (10) as λ (ηt −1 , ~pt ) , we havept +1  − γA a(1 − δ )( ~
d~
p) 2
=
 dη  − (1 − δ )a t  Two eigenvalues of the Jacobian matrix are −of (12) leads to (1 − δ )( ~p)2 =satisfies −γA(1 − a~p )
a

pt 
 d~
0


.
∂λ (η , ~p ) ∂η  dηt −1 

(15)

γA
and ∂λ (η , ~p ) ∂η . Rearrangement2~a(1 − δ ) p. One eigenvalue

1
γA=−< −1 , since the above equality requires 0 < ~p < 1 a if ~p2~~
1 − apa(1 − δ ) p

should be positively valued.

p ) ∂η = −Next we examine another eigenvalue ∂λ (η , ~s (1 − γ )aA(1 − δ ). Rearranging (13)(η ) 2p}aη . It is clear that
leads to η 2 − s (1 − γ )aA(1 − δ ) = {s (1 − γ ) A − (1 − δ ) ~η 2 > s (1 − γ )aA(1 − δ ) holds if (1 − δ ) ~p < s(1 − γ ) A is met. On the other hand, we obtain7

the equality of (14) from (12). If (14) is met, − 1 < ∂λ (η , ~p ) ∂η < 1 holds.
One eigenvalue is greater and another is smaller than unity in absolute value. Given the initialcondition η −1 , there exists a unique equilibrium path that converges to the balanced-growthpath. Q.E.D.

Given that the price of old capital is constant over time, the equilibrium growth rate ofinvestment converges to the balanced-growth path through endogenous fluctuations. A greater
investment in new capital of the current period implies a greater stock of old capital purchasedby agents of the next generation at the end of the subsequent period, which in turn tends tocrowd out new capital investment made by the next generation. The counter-cyclical behavior
between new capital and old capital is the driving force of endogenous fluctuations of economic
growth.
Figure 1-1 and 1-2 illustrate the dynamic behavior of growth rates of investment, capital,and consumption, each of which converges to the balanced growth path through endogenous
fluctuations. Capital and investment move in the same direction, but consumption and capital(or investment) move in the opposite direction. One can intuitively see this fromCt ≡ Cty + Cto = (Wt − St ) + {(1 - δ ) pt I t −1 + γAK t }= {(1 − γ ) AK t − ( I t + (1 − δ ) pt I t −1 )} + {(1 - δ ) pI t −1 + γAK t } = AK t − I t .

(16)

Given output AK t , the aggregate consumption, Ct , is inversely related to investment in newcapital investment.

4. Technological Change
In the 1990s the U.S. economy resurged along with the development of information
technology (IT). The new technology promoted broader applications in computing andcommunications, and drastically changed the production process and life style. This
technological revolution did not diffuse equally among all firms, but favored newly establishedfirms over incumbent ones [e.g. Hobjin and Jovanovic (2001)]. In order to analyze thisphenomenon, we consider a situation in which technological progress is embodied only in
newly invested capital. Assume that at the beginning of period T people experience an
8unanticipated rise in the productivity of newly invested capital. We assume thataT +1 = aT + 2 =,..., = a** > aT = a * . Notice that the productivity of the existing old capital is notaffected by the technological change. For later reference, we denote steady state values as

p (a) and η (a) , respectively.
depending on the technological parameter, such that ~p (a * ), η (a * )} ,Until period T-1, the steady state equilibrium is described as a pair { ~satisfyingγA
a *{γA + (1 − δ ) ~p (a * )} = ~ * ,p (a )andη (a * ) = s(1 − γ ) A(a * + a *1−δ) − (1 − δ )a * ~p (a * ) .
*η (a )(17)(18)

The technological revolution occurs at period T, and equilibrium conditions are described asγAa **{γA + (1 − δ ) ~pT +1} = ~ ,
pT

(19)

and

η T = s (1 − γ ) A(a * + a *1−δ) − (1 − δ )a * ~pT .
*η (a )(20)

Note that the new productivity value a ** appears only in (19). If, instead, the technologicalchange would diffuse equally between new and old capital, the second term in the R.H.S. of(20) should be replaced by (1 − δ )a** ~pT . We summarize interesting findings in the following.

Result 1
Assume that there is an unanticipated technological change at the beginning of period T. The
price of old capital measured in efficiency unit ~pT declines. The price of old capital measuredpT ) also declines.
in the physical unit pT ( ≡ aT ~pT +1 = ~p (a** ) follows.
Proof: People anticipate the productivity change to be permanent, and ~pT = ~
p (a** ) through arbitrage. Additionally,
The backward induction of expectation leads to ~p (a) is decreasing in a . The first part follows from this. The vintage-specificas seen in (14), ~parameter of old capital remains a * , and the price of old capital measured in physical unitp (a ** ) that is smaller than a * ~p (a * ) . The second part follows from this. Q.E.D.
becomes a * ~The rise in productivity is embodied only in new capital, but not in old capital, and so agents9

invest more in new capital and less in old capital, followed by a fall in the price of old capital.

Result 2
Assume that there is an unanticipated technological change at the beginning of period T. New
capital investment grows faster than otherwise.
Proof:

Since ~
pT −1 = ~p (a * ) > ~pT = ~p (a ** ) , comparing (18) and (20) leads to ηT > ηT −1 .

Q.E.D.

The fall in the old capital allows the greater amount of saving channeled into investment in newcapital. The “substitution effect” reinforces investment in new capital.
After period T+1, a rise in productivity is embodied in all the existing capital, and the priceof old capital is expected to rise again. We obtain the following.

Result 3
Assume that there is an unanticipated technological change at the beginning of period T. The
price of old capital measured in physical unit rises from period T+1 on.
Proof:

p (a)} that
It follows from (19) and the definition of p(a ){≡ a~(1 − δ ){ p(a)}2 + aγAp (a ) − aγA = 0 . Differentiating the above equation leads todp(a)γA{1 − p(a)}=> 0 , since the rearrangement of the above equation impliesda2(1 − δ ) p (a ) + aγA(1 − δ ){ p(a)}2 = aγA{1 − p(a)} which requires 0 < p(a ) < 1 , if p(a) should be positively
valued. Q.E.D.

From period T+1 on, the no-arbitrage condition are described as (19). At period T+1, the market
clearing condition in the good market is given byη T +1 = s(1 − γ ) A(a ** + a *

1−δ

ηT

) − (1 − δ )a ** ~p (a ** ) .

(21)Comparing (20) and (21), we find that an increase in a has two opposite effects on the growthrate of new capital investment. On the one hand, the improvement in the efficiency of capitalincreases output and saving, while on the other hand, it also increases the value of old capital.

10

The former tends to promote investment in new capital, whereas the latter tends to shrink it. The
total effect on η T +1 is ambiguous. From period T+2 on, the market clearing condition in thegood market is given byη T +i +1 = s(1 − γ ) A(a ** + a **1−δ

η T +i) − (1 − δ )a ** ~p (a ** ) i = 1,2,3,... .

(22)

We next turn to the analysis of an anticipated technological change. The emergence of theinternet, although there were initially only a few internet-based investment opportunities, in theeyes of many observers the changes signaled fast future productivity growth and an increased
potential for obsolesce of existing capital. Before the upcoming introduction of a newtechnology, news about its arrival will affect investment and the price of the existing capital.
We study the simplest case in which technological change that comes at T is forecasted one
period in advance. We examine the effect of the anticipated technological change by assumingthat at the beginning of period T-1 people anticipates the arrival of technological change
anticipates at T. We summarize main properties in the following two results

Result 4
Assume that the arrival of technological change at period T is anticipated at the beginning of

pT −1 rises. The price of oldperiod T-1. The price of old capital measured in efficiency unit ~capital measured in the physical unit pT −1 rises. After period T, prices of old capital measured
in efficiency and physical units behave in the same manner as the case of the unanticipatedshock.
Proof:

We first examine the third part. Since period T when the information about thetechnological shock is revealed, it is straightforward to see that the behavior of old price inefficiency unit is the same between the anticipated and the unanticipated changes. No-arbitrage
condition at period T requires

γA + (1 − δ ) ~pT +1 =

γA

(23).

a ~
pT
**

If people anticipate the steady state price of old capital to be sustained from period T+1 on, thepT = ~
pT +1 = ... ≡ ~p (a ** ) follows to satisfy (23).
equality ~When the technological change is anticipated one period in advance, people at period T-1

11invest in assets by conjecturing ~pT = ~p (a ** ) . No-arbitrage condition at T-1 requiresγA
a**{γA + (1 − δ ) ~pT } = ~
pT −1

(24)

p (a ** ) satisfies
Of course, ~γA
a**{γA + (1 − δ ) ~p (a** )} = ~ ** .
p (a )p =~p (a ** ) that
It follows from (30), (31), ~

(25)

T

pT −1 ≡ a * ~pT −1 = a ** ~p (a ** ) ≡ p(a ** ) ,(26)

and with a * < a ** and Result 3.

p (a * ) < pT −1 = p(a ** )(27)

follows. The first part follows from this.

p (a* ) < ~pT −1 follows since p(a * ) = a * ~p (a * ) < a * ~pT −1 ≡ pT −1 follows from
The inequality ~equations (26), (27), and a * < a ** . The second part follows from this. Q.E.D.

In the case of an anticipated shock, the stock market reacts before the shock actually arrives.
Result 4 suggests that the path of investment in new capital will differ between an anticipatedcase and an unanticipated case. We obtain the following.

Result 5
If the arrival of technological change at period T is anticipated one period in advance, thegrowth rate of new capita investment is smaller at period T-1 and greater at period T thanotherwise.
Proof: First we show the former part of the Result. The growth rates of investment at T-1 ηTA−1and ηTU−1 satisfy,η TA−1 = s(1 − γ ) A{a * + (1 − δ )η TU−1 = s(1 − γ ) A{a * + (1 − δ )a*η TA− 2a*

η TU− 2} − (1 − δ ) pTA−1(28)} − (1 − δ ) pTU−1(29)

where the superscripts A and U denote the case of anticipated technological shock andunanticipated one, respectively. Since the economy is in the steady state until t-2,

12

η TU− 2 = η TA− 2 (= η * ) , Result 3 shows that pTU−1 = p(a * ) < p(a ** ) = pTA−1 , thenη TA−1 < η TU−1 (= η * ) . Now we show η TA > η TU .Growth rate of investment at period T is
described byη TA = s(1 − γ ) A{a * + (1 − δ )a*ηη TU = s(1 − γ ) A{a * + (1 − δ )

A
T −1
*

a

η

U
T −1

} − (1 − δ ) pTA(30)} − (1 − δ ) pTU(31)p (a ** )) and η TA−1 < η TU−1 (= η * ) .
We obtain η TA > η TU since pTU = pTA (= a * ~Q.E.D.

When the technological change arrives, there occurs a greater investment boom if it isanticipated beforehand than not. In the case of anticipated technological change, young agents atperiod T-1 foresee that technological progress at period T makes capital newly invested atperiod T-1 obsolete and its valuation low. They find new capital investment at T-1 less attractive
and hence old capital more attractive even there are no productivity shocks at T-1. The
consequent smaller market valuation of old capital at period T allows a greater amount of saving
channeled into investment in new capital than the case of an unanticipated case.
It is useful to review the net national product (NNP) in the present model. We define the net
national product (NNP) asγA
Wt + I t −1{γAat + (1 − δ )at ~pt − 1} + (1 − δ )at −1 ~pt −1I t − 2 ( ~ − 1) ,
pt −1

(32)

where the first term represents the labor income and the second and third terms the net income
from new and old capital, respectively. On the other hand, the depreciation of capital comprises
two components, such that

{1 − (1 − δ )at ~pt }I t −1 + (1 − δ )at −1 ~pt −1 I t −2 .

(33)

The first term represents the depreciation of new capital and the second term the depreciation ofold capital. Interestingly, new capital depreciates in two ways, first through the physicaldepreciation captured by δ and second through the economic depreciation represented by thefall in ~pt . It is this aspect of capital depreciation that Laitner and Stolyarov (2003) concentrateon in constructing Tobin’s Q. Equations (32) and (33) imply that the gross national product(GNP) is Yt . It follows from Result 1, (32), and (33) that, when the technological change

13arrives, capital depreciates faster, NNP declines, but GNP does not change.
The market value and the book value of capital are equal because the existing capital isnecessarily traded in the secondary market in our model with two-period-lived agents. However,if the model is extended to allow for longer-lived agents or firms, the firms may keep holding
the existing capital. If firms do not revise asset values of capital downward in response to thetechnological change, depreciation of capital would finally be undervalued on national accounts.
Finally in this section we examine the impact of technological change on welfare. We focus
on the welfare analysis in terms of the possible conflict of interest among different generations.8
In the face of the unanticipated shock, we calculate the lifetime consumption of each generation.
The young-age consumption of the generation t is proportional to wage income, and is

Cty =

β
1+ β

Wt =

(1 − γ ) βA
{at I t −1 + (1 − δ )at −1 I t −2 } .
1+ β

(34)

The old-age consumption is

C to = γA{at +1 I t + (1 − δ )at I t −1 } + (1 - δ )a t +1 ~pt +1 I t .

(35)

The first term represents the rent from capital, and the second term the gain from selling capitalto the subsequent generation. In a growing economy, the benefits of technological change are
greater over time. However, Result 1 suggests that people of some generation incurs capital
losses from holding old capital and may be worse off.
We examine the lifetime consumption of the two generations who are alive when thetechnological change arrives. For agents born at T, the young-age consumption remains
unchanged, but the old-age consumption changes to

CTo+1 = γA{a ** I T + (1 − δ )a * I T −1} + (1 - δ )a ** ~p (a ** ) I T .
(36)p (a)} rises (Result 3). The young
When a rises, I increases (Result 2) and p (a ){≡ a~T

generation is better off. On the other hand, for agents born at T-1, the young-age consumption
remains unchanged, but the old-age consumption changes to

CTo = γA{a * I T −1 + (1 − δ )a * I T −2 } + (1 - δ )a * ~p (a ** ) I T −1 .

(37)

In the face of the unanticipated technological progress, the old generation finds the decline in8

While a number of works recently argue the effect of technological progress on the welfare interms of increasing income inequality between skilled and unskilled workers (e.g. Jovanovic (1998)
and Caselli (1999)).
14

the price of old capital to fall (Result 1). The old generation is worse off.
It is straightforward to see that people that are born from period T+1 on are better off. Old
agents incur capital losses by holding old capital because technological progress leads toeconomic obsolescence in old capital. This finding suggests that when a rise in productivity is
embodied only in new capital, there may be a conflict of interest between people who enjoy thebenefits of holding new capital and who incur capital losses from holding old capital.
In the case of the anticipated technological change, the conflict of interest arises in the
different way. Briefly speaking, when the technological change is anticipated at T-1, it affects

pT −1 as well pT . Since the rise in price of old capital at T-1 increases the consumption levelof generation T-2 and decreases that of generation T-1, lump-sum transfer between generation
T-2 and T-1 is also needed for Pareto Improvement allocation.
If the transfer of income is permitted through government intervention, however, the welfare
of agents born at both periods T-1 and T may be improved. Intuitively, an increase inproductivity from period T+1 on will lead to an increase in output after period T+1 while output
of before period T remains unchanged. We obtain the following.

Proposition 2
When there is technological change, there exists a lump-sum transfer of income that implementsa Pareto improvement.
Proof: We mainly prove the case of unanticipated technological change, and briefly state on the
anticipated case. As in the proof of Result 5, the superscripts A and U denote the case of
anticipated technological shock and unanticipated one, respectively. Variables without
superscripts denote the case without any technological change.
First we show that the output at period T remains the same as that when the technologicalchange does not happen. The output at period T, YT , is shown to be

YTU = AK TU = Aa *{I TU−1 + (1 − δ ) I TU−2 } = CTYU + CToU + I TU
Since I TU−1 and I TU− 2 are not affected by the technological change, YTU = YT . Since

YTU = YT = CTY + CTo + I T , an allocation CTYU = CTY , CTOY = CTO , and I TU = I T can berealized by an appropriate transfer scheme. Second we show that the consumption level of15

U
generation T at period T+1, CTOU
+1 can be increased. The total output at period T+1, YT +1 , is

YTU+1 = AK TU+1 = A{a ** I TU + (1 − δ )a * I TU−1} = CTYU+1 + CToU+1 + I TU+1 , which is greater thanYT +1 Therefore, an appropriate transfer by government can achieve Pareto-improving allocation.
In the case of an anticipated technological change, an additional transfer is needed at period
T-1, since the change affects pTA−1 as well pTA . At period T-1, a transfer similar to one
described above can set CTYA−1 = CTY−1 , CTOA−1 = CTO−1 , and I TA−1 = I T −1 . The same transfer
scheme at period T as one in the unanticipated case achieve Pareto-improving allocation. Q.E.D.

5. Oscillatory Equilibria
In this section we study the possibility of a regime switch from an economy with both newcapital and old capital active to an economy with only one type of capital active.
If the equilibrium switches to an economy in which only new capital is in operation, oldcapital would become completely obsolete. However, this is impossible so long as δ < 1 .
Hence we focus on a regime change in which the equilibrium switches to an economy in which
no new capital investment occurs and only old capital is in operation. Intuitively, this regime
switch involves an appreciation in old capital and crowding out of new capital investment.
Throughout this section we assume that the vintage-specific parameter is constant over time,such that a ≡ at = at +1 = ... . Assume that there is a break period T (0 ≤ T < ∞ ) when newcapital investment is not made. Until period T-1, then any equilibrium sequence remains to be
described by (10) and (11).
Since all the saving should be channeled to the purchase of old capital at period T, themarket clearing in the good market requires S T = (1 − δ ) pT I T −1 since I T = 0 . By
incorporating ηT = 0 into (10), we obtains (1 − γ ) A(1 +1−δη T −1) = (1 − δ ) ~
pT .

(38)

old
If equilibria satisfy (38), agents have to prefer old capital to new capital to satisfy RTnew+1 ≤ RT +1 ,or equivalently

γA
γA + (1 − δ ) ~pT +1 ≤ ~ .
ap
T

16

(39)

The asset price at the break period ~pT should be sufficiently high to satisfy (38) so that thepT +1 should be sufficiently low to satisfy the inequality (39).
anticipated price ~We need some elaboration to describe the equilibrium at period T+1 because I T = 0
implies that the secondary market for old capital is not opened at period T+1. At period T+1
investment in new capital absorbs all the saving. The equilibrium condition is simply given bys(1 − γ ) AK T +1 = I T +1 .

(40)

Note that K T +1 = a (1 − δ ) I T −1 ; the outstanding stock of capital at period T+1 succeeds to only
capital that has originally been invested at period T-1. It is impossible to write the equilibrium
condition at period T+1 in terms of ηt since I T = 0 . If both new and old capital are held fromT+2 on, (10) and (11) again describe equilibrium conditions. Here we characterize equilibria
more completely by allowing for the possibility of the regime change.

Definition 2
Assume that the technological parameter at is constant over time. (I) When T → ∞ , any
perfect-foresight equilibrium is a sequence of variables {ηt , ~pt }t∞=0 satisfies (10), (11) withequality, given η −1 > 0 , for any t . (II) When there is a T < ∞ , any perfect-foresightequilibrium is a sequence of variables {ηt , ~pt }t∞=0 satisfies (10), (39) with equality for any

t except for t = T , T + 1 , satisfies (38), (39), and ηT = 0 for t = T , and ηT +1 → ∞ fort = T + 1 , given η −1 > 0 .
Figure 2 illustrates the dynamics of the price of old capital in the ( ~pt , ~
pt +1 ) plane. Any ofp oscillates. This figure shows that at
price paths that lie apart from the steady state price ~period T the wealth constraint (10) binds with ηT = 0 and is replaced by (38). At this moment
the no-arbitrage condition can be violated.
Since the secondary market for capital does not exist at period T+1, the backward inductionof expectation formation does not work between T and T+1. It turns out that the behavior of theeconomy before the break period T is not constrained by the behavior after T. On regime switch,what connects the economies before and after the break period T is only the outstanding capital

17

stock K T +1 that is inherited from investment in period T-1 to satisfy K T +1 = (1 − δ ) aI T −1 , andhence the violation of the no arbitrage condition can be consistent with rational behavior ofagents. Given the “new” initial condition KT +1 , the economy restarts. A new sequence of theprice of old capital { ~pt }tS=T + 2 ( S ≤ ∞) is defined irrespective of the past sequence, showing
either oscillatory path again or a convergence to the steady state.
Given the same parameter values and initial conditions, the equilibrium can exhibit either aconvergence to the balanced-growth path or oscillatory equilibria, depending on the investors’
expectations. If agents anticipate a constant price of old capital over time, the self-fulfilling
expectation dictates the equilibrium to converge to the balanced-growth path. However, if
agents anticipate the stock market boom, the equilibrium exhibits an oscillatory behavior.
Although the price of old capital exhibits bubble-like behavior, the asset value reflects only
economic fundamentals, but does not include the “rational bubbles” defined by Tirole (1985).
The standard overlapping-generation models with capital that depreciates at constant rate
excludes explosive paths as violating the no-arbitrage condition at the instant when bubbles
absorb all the savings and hence are inconsistent with rational behaviors of investors if bubbles
should be held.9 The framework with finitely lived agents and capital, however, allows paths
that show the bubble-like behavior of asset prices to be consistent with perfect-foresightequilibria. Since capital survives only a finite period and hence the composition of capitalchanges over time, the aggregate market value of capital need not continue to grow faster overtime.
We illustrate the behavior of an oscillatory equilibrium. Before doing so, we have to checksome properties of oscillatory equilibria. Note that the inequality (39) may restrict the parameter
space under which oscillatory equilibria exist. We have the following.

Result 6
A necessary condition for the existence of any oscillatory equilibria is s (1 − γ )aA < 1 − δ . The
9

If an asset in question should be rolled over forever, any equilibrium with explosive bubbles could
be sustainable only when the asset could be purchased at higher prices by successive generations thatearn even higher wage income. This is impossible since then the market value of the bubble should
exceed the aggregate savings.
18

price of old capital at the break period T, pT , must satisfys(1 − γ ) A< pT ≤ 1 a .
1−δ

11γA( ~ − 1) . Since any expected price
apT
1−δ
~
pT ≤ 1 a follows. Rearranging (38) yields
pTe +1 should be nonnegative, the inequality ~pTe +1 ≤Proof: Rearranging (39), we obtain ~s(1 − γ ) A(1 − δ )η T −1positive, andI= (1 − δ ) ~pT − s(1 − γ ) A . The initial condition η T −1 ≡ T −1 > 0 has to be
I T −2s(1 − γ ) A ~pT ≤ 1 a jointly imply that
< pT has to be met. This inequality and ~1−δs (1 − γ )aA < 1 − δ is a necessary condition for any oscillatory equilibria to be viable. The
latter inequalities are immediate. Q.E.D.

In addition, we have more restrictions on parameter values according to when the break period
arrives.

Result 7
Assume that s (1 − γ )aA < 1 − δ holds. A oscillatory equilibrium with T=0 exists if

η −1 ≥s (1 − γ )aA(1 − δ ). A oscillatory equilibrium with T=1 exists
1 − s (1 − γ )aA
 γ (1 − δ )s(1 − γ )(1 − δ )+− s(1 − γ ) .
1 − δ + aγA 1 − δ − s (1 − γ )aA


if 0 < η −1 ≤ s (1 − γ )(1 − δ ) Proof: We first prove the earlier part. Result 6 reveals that, in any oscillatory equilibrium,s(1 − γ ) A ~< p 0 ≤ 1 / a must hold at T=0. We wish to prove that such an equilibrium can be1−δs (1 − γ )aA(1 − δ ).
constructed if η −1 ≥1 − s (1 − γ )aAs (1 − γ ) A[1 + (1 − δ ) / η ] . It is easy to show that1−δs (1 − γ ) A
s (1 − γ )aA(1 − δ )Γ(η ) =as η → ∞ , and Γ(1 / a ) =. Since Γ(η ) is a
1 − δ − s (1 − γ )aA
1−δ
Define a function Γ(η ) ≡monotone decreasing function,η≥s (1 − γ ) A< Γ(η ) ≤ 1 / a is satisfied if
1−δs (1 − γ )aA(1 − δ )holds. Hence, an equilibrium satisfying
1 − δ − s (1 − γ )aA19s (1 − γ ) A ~s (1 − γ ) A[1 + (1 − δ ) / η −1 ] = (1 − δ ) ~p 0 and< p 0 ≤ 1 / a can be constructed if
1−δs (1 − γ )aA(1 − δ )η −1 ≥is met.
1 − δ − s (1 − γ )aAs (1 − γ ) A ~< p1 ≤ 1 / a must hold in
1−δγA(1 − δ )p0 =from (11) andthe same way as the earlier part. We obtain (1 − δ )a~p1 + γA(1 − δ ) ~Second, we prove the latter part. We can show thatη0 =s (1 − γ ) A(1 − δ )from (38), and so (1 − δ )a~p 0 + η 0 is decreasing in ~p1 , taking~(1 − δ ) p1 − s (1 − γ ) A

s (1 − γ ) A(1 − δ )γA(1 − δ )+, ∞) . Equation (10) must hold at t=0,
(1 − δ ) / a + γA (1 − δ ) / a − s (1 − γ ) A
s (1 − γ )aA(1 − δ ). Since (1 − δ )a~p 0 + η 0 has theand rearranging it yields η −1 =~(1 − δ )ap0 + η 0 − s (1 − γ )aA
the value within [range described above, we obtain the range of η−1 as γ (1 − δ )s (1 − γ )(1 − δ )+− s (1 − γ )] . If η −1 lies in this(0, s (1 − γ )(1 − δ ) 1 − δ + aγA 1 − δ − s (1 − γ )aA~~p ≥ 0,
interval, we can construct an equilibrium defined by {η , p , p } satisfying η > 0 , ~
0

and

0

1

0

0

s (1 − γ ) A ~< p1 ≤ 1 / a . Otherwise such an equilibrium cannot be constructed. Q.E.D.
1−δ

It is easy to construct an oscillatory equilibrium with the break period at T=1. The goodmarket clearing condition at period t = 0 becomes

η 0 = s(1 − γ ) Aa(1 +

1−δ

η −1

) − (1 − δ )a~
p0 .

(41)

The no-arbitrage condition at period 0 implies

γA(1 − δ ) ~p1 + γA = ~ .
ap0

(42)

Finally, the good market clearing condition at the break period (T=1) impliess (1 − γ ) A(1 +

1−δ

η0

)= ~
p1 .

p0 , ~
p1 , η0 } , as long as theEquations (41), (42), and (43) yield an equilibrium sequence, { ~condition presented in Result 6 is met.
We conduct a numerical analysis. We set A = 2.6 , a = 1, γ =.3 , δ = 0 , and

20

(43)

β = .99 20 = 0.817 . We take twenty years for one period. Additionally, we set I − 2 = 1 andI −1 = 0.186 , and so η −1 = 0.186 . The steady-state pair { ~p * ,η * } is {0.575, 1.034}.10 Figure
3-1 through 3-5 illustrates the dynamic behavior over the periods covering before and after the
break period. In all the figures the “white“ dot illustrates the convergent path with ~pt = ~
p * forall t’s, and the “black” dot an oscillatory path.

Period 0: Outset of the belief in the boom
Agents, based on a “sunspot”, believe that the stock market boom will happen next period. At
the current period agents anticipate the smaller price of old capital than unless the boom is

p 0 = 0.441 < ~p * =0.575. the smaller market
anticipated. As shown in Figure 3-1, we have ~value of old capital enables more saving to be channeled into investment in new capital (Figure3-4). The stock market capitalization relative to GNP, p 0 I −1 / Y−1 , is smaller than the one in theconvergent path (Figure 3-2).

Period 1: Break Period: Boom in the Stock Market
The price of old capital rises (Figure 3-1), and accordingly the stock market capitalization
relative to GNP rises (Figure 3-2), accompanied by zero investment in new capital (Figure 3-4)
and the greater consumption (Figure 3-5). The stock market boom is associated with
consumption boom.

Period 2: Recession
The stock market capitalization relative to GNP becomes zero because the secondary market
for old capital is not opened. The level of output drops sharply, as captured by a negative growthrate of output (Figure 3-3). The decline in output arises from the smaller available stock( K 2 = I 0 < K1 = I 0 + I −1 ), which arises because any investment in new capital has not been
made at period 1. Consumption also declines (Figure 3-5). All the saving is channeled to10

If we take one period as twenty years, we get η −11 / 20 = 0.919 . Then the average annual growthrate is –8.1%. Similarly, the steady-state annual growth rate corresponding to η is 0.2%..
*

21

investment in new capital and investment grows faster. Note that the growth rate of investment
in new capita is infinite although it is not plotted in Figure 3-4.

Period 3: Bottom
Based on a “sunspot”, agents recur to forming a “normal” expectation. The old capital isquoted at the steady-state price p * =0.575 (Figure 3-1). Surprisingly, the output level is evensmaller than the previous period, as captured by a negative growth rate of output (Figure 3-3).
This reflects the fact that newly invested capital at the previous period only accounts for theoutstanding stock of capital. We have K 3 = I 2 that is smaller than K 2 = I 0 .

Period 4: Symptom of Recovery
The economy begins to recover from the recession and converge to the balanced-growth paththrough endogenous fluctuations (Figure 3-5).

Noteworthy, a persistent recession comes after the stock market boom ends. The stock market
boom occurs at the sacrifice of investment in new capital. The available stock of capital declines
over a number of periods, and prolongs the recession.

6: Conclusion
We construct a growth model of overlapping generations based on vintage capital and derive
several findings that could not be obtained otherwise. When investment-specific technologicalchange arrives, the economy converges to a new balanced growth path with a higher growth rateof output through endogenous fluctuations, but stock market capitalization, although eventhough it finally grows, initially shrinks to the extent that new capital producers do not go public.
Technological change leads to faster depreciation of capital.
We find a different prediction from the standard theory about the behavior of the stock
market and its interaction with economic fundamentals that drive fluctuations in the economy.
The stock market may show perverse behavior as a predictor of the future economic
fundamentals. This is because the market for new capital is often “missing”. Although the22technological revolution leads to the rise in the relative price of new to old capital, we may
observe only the devaluation of old capital in the public market. Additionally, as we showed in
the analysis of oscillatory equilibria, we may misread the rise in the relative price as a future
economic boom.
Future research may take several directions. One important task is to exploit consequencesof faster depreciation of existing capital when technological change arrives.11 As argued in
Section 4, if firms keep holding old capital, the national account may underestimate
depreciation of capital and overestimate NNP. This theoretical finding gives an importantsuggestion on how to evaluate the existing capital when technological changes occur. We mightobtain a similar result on the contribution of the investment-specific technological change on
economic growth as that of Greenwood et al (1997).
An analysis of the impacts of divergence between the market value and book value ofexisting capital held by incumbent firms may give a hint to explain the conservative behavior ofthese firms relative to new firms [e.g. Hobjin and Jovanovic (2001)]. Assume that the model isextended to allow for longer-lived agents or firms. When the technological change arrives, firms
will attempt to adjust the book value of the capital only slowly relative to market evaluation andmay undervalue the asset value of new investment.
Throughout this paper we have focused on the effect of exogenous technological change. It
is interesting to introduce the process of technology adoption into the model. One way is tointroduce in some way the notion of the match between new technology and skilled workers
into the model. Another way is to apply the possible conflict of interest among different
generations regarding the arrival of new technology, as studied in the last part of Section 4. If
the young take the initiative in adopting new technology, the economy will tend to adopt newtechnology, whereas if the old have the initiative, the economy may not adopt it. If new capitalis produced by a the subsidiary of a company whose main office may be described by theaggregate production function, the age distribution of the company may influence the choice of
new technology.
We have assumed that young capital and old capital are perfect substitutes. It is of interest to
11

See the important work by Laitner and Stolyarov (2003).
23

extend the model to allow for imperfect substitutability of different vintages of capital. The
technological revolution may be interpreted as an extreme case when the elasticity of
substitution becomes infinity. The more general framework may be useful to make for a richer
analysis.
Government policies may perversely influence economic growth. Tax reform, such as thedividend tax reduction advocated by the Bush administration, can increase the price of oldcapital, thereby deterring economic growth.
One interesting theoretical topic for investigation is whether the stationary bubbles definedby Tirole can exist in our model. Weil (1990) demonstrates that under production technology
similar to ours, the stationary bubbles will exist. Furthermore, the introduction of the notion of
vintage capital into the original Diamond model is an important extension. Although the
non-linear property of the Diamond model may make the analysis challenging, this line ofinvestigation is worth pursuing.

24

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26

Figure 1-1: Evolution of η t = I t

0

1

2

3

Figure 1-2: Evolutions of4

I t −1

5

6

7

8

9

K t / K t −1 and C t / C t −1

CapitalConsumption

0

1

2

3

4

5

6

7

27

8

9

Figure 2

Oscillatory Price Path

~
pt +1

1a

s(1−γ )A(1+

1

ηT−1

)

~
p

0

~
p ~
pT −1 ~
pT −2 ~
pT

29

~
pt

Figure 3-1
pt (= pt )Price of Old Capital: ~
1.2
1
0.8
0.6
0.4
0.2
0
0

1

2

3

4

Oscillatory economy
Convergent economy
The price ~p 2 is not defined for the oscillatory economy since there is no old capital atperiod 2.

Figure 3-2
Market Capitalization/GNP
0.3
0.25
0.2
0.15
0.1
0.05
0
0

1

2

29

3

4

Figure 3-3
Annual Growth Rate of Output
1.5
1
0.5
0
0

1

2

3

4

-0.5
-1
-1.5

Figure 3-4
Annual Growth Rate of New Capital investment
10
8
6
4
2
0
-2

0

1

2

3

4

-4
-6
-8

Growth rates of new capital investment at periods 1 and 2 are not defined for theoscillatory economy.

30

Figure 3-5:
Annual Growth Rate of Consumption
1.5
1
0.5
0
-0.5

0

1

2

-1
-1.5
-2
-2.5
-3
-3.5

31

3

4