We highlight the salient characteristics and implications of the seminal contributions in the field of vintage capital growth theory (proposed entry for the new Palgrave dictionary of economics, 2nd edition)Obsolescence; embodiment; technology diffusion; vintage human capital

David, DE LA CROIX

Omar, LICANDRO

Raouf, BOUCEKKINE

English

Research Papers in Economics

Département des Sciences Économiques
de l'Université catholique de Louvain
Vintage Capital
R. Boucekkine, D. de la Croix and O. Licandro
Discussion  Paper   2006-14Vintage Capital
Raouf Boucekkine
Department of Economics and CORE, Universit´ e catholique de Louvain.
David de la Croix
Department of Economics and CORE, Universit´ e catholique de Louvain.
Omar Licandro
European University Institute, Florence.
Abstract
We highlight the salient characteristics and implications of the seminal con-
tributions in the ﬁeld of vintage capital growth theory (proposed entry for the
new Palgrave dictionary of economics, 2nd edition).
Keywords: Obsolescence, embodiment, technology diﬀusion, vintage human
capital.
JEL Classiﬁcation Numbers: O41, E22, E32, J24.
CORE DP 2006/24
1In the Neoclassical growth theory capital is assumed homogeneous and technical
progress disembodied, meaning that all capital units equally beneﬁt from any tech-
nological improvement. The disembodied nature of technical progress looks barely
unrealistic, as acknowledged by Solow (1960, p 91): “...This conﬂicts with the casual
observation that many if not most innovations need to be embodied in new kinds of
durable equipment before they can be made eﬀective. Improvements in technology af-
fect output only to the extent that they are carried into practice either by net capital
formation or by the replacement of old-fashioned equipment by the latest models...”
Accounting for the age distribution of capital is a way to cope with this criticism, and
this actually suggested an important stream of the growth literature of the 50’s and
60’s, giving birth to the vintage capital theory.
An economy is said to have a vintage capital structure if machines and equipment
belonging to separate generations have diﬀerent productivity –or face diﬀerent depre-
ciation schedules as in Benhabib and Rustichini (1991). Let us denote by I (v) the
number of machines of vintage v. With zero physical depreciation, vintage technology
v is
Y (v,t) = F (I(v),L(v,t),e
γv),
where L(v,t) is the amount of labor assigned to this vintage at time t ≥ v. Parameter
γ > 0 designates the rate of technical progress, which is said to be embodied since it
only beneﬁts vintage v. F(.) has the properties of a neoclassical production function.
Vintages produce the same ﬁnal good
Y (t) =
Z t
t−T(t)
Y (v,t) dv,
where Y (t) is total production and T (t) is the lifetime of the oldest operative vintage.
The Lifetime of capital. In Johansen (1959), technical progress is labor-saving
and technology putty-clay, meaning that capital-labor substitution is permitted ex-
ante, but not once capital is installed. Because factor proportions are ﬁxed ex-post,
Y (v,t) = F (I(v),e
γvL(v,t)) = g (λ(v)) I(v),
where the labor-capital ratio λ(v) and the size of the capital stock I(v) are both
decided at the time of installation, and employment is L(v,t) = λ(v)eγvI (v).
In Johansen, obsolescence determines the range of active vintages. Quasi-rents of
2vintage v at date t are proportional to g (λ(v)) − λ(v) eγv w(t), where w(t) is the
equilibrium wage. Since wages are permanently growing, as a direct consequence of
technical progress, quasi-rents are decreasing. Machines of vintage v are operated as
long as their quasi-rents remain positive. Consequently, the scrapping age is deﬁned
by T = t∗−v where g (λ(v)) = λ(v) eγvw(t∗). Therefore, Johansen’s framework leads
to an endogenous, ﬁnite lifetime of capital.
The Embodied Question. In Solow (1960), vintage technology is Cobb-Douglas
Y (v,t) = [e
γvI(v)]
1−α L(v,t)
α,
and the capital-labor ratio adjusts continuously. The embodiment hypothesis takes
the form of quality adjustments, with capital’s quality growing at rate γ. In sharp
contrast to Johansen, capital lifetime needs not be ﬁnite, since under Cobb-Douglas
technology any wage cost could be covered by assigning arbitrary small amounts of
labor.
A striking outcome of Solow’s model is its aggregation properties. Denote by L(t) the
total labor supply, and deﬁne quality adjusted capital as
K(t) =
Z t
−∞
e
γvI(v) dv. (1)
Since marginal labor productivity equalizes across vintages, aggregate output becomes
Y (t) = K(t)
1−αL(t)
α.
Aggregate vintage technology in Solow (1960) degenerates into a neoclassical produc-
tion function. However, by diﬀerentiating (1), the motion law for capital is slightly
diﬀerent
K
′(t) = e
γt I(t)
reﬂecting embodied technical change. Since e−γt measures the relative price of invest-
ment goods at equilibrium, the value of capital is by deﬁnition A(t) = e−γt K(t), and
evolves following
A
′(t) = I(t) − γ A(t).
Technological progress operates as a steady improvement in equipment quality, which
in turn implies obsolescence of the previously installed capital. In Solow, obsoles-
3cence does not show up through ﬁnite time scrapping but through labor reallocation
reﬂecting a declining value of capital.
This important point has been at the heart of a recent literature on the productivity
slowdown and the information technology revolution (see Whelan, 2002). Actually,
the potential implications for growth of embodied technical progress was tremendously
controversial in the 60s. In a famous statement, Denison (1964) claimed “the embod-
ied question is unimportant.” His argument was merely quantitative and restricted
embodiment to changes in the average age of capital in a one-sector growth accounting
exercise. In particular, his reasoning omits de facto the relative price of capital chan-
nel. Greenwood, Hercowitz and Krusell (1997), by using Gordon (1990)’s estimates
of the relative price of equipment, quantitatively evaluate the Solow model, claim-
ing that around 60% of US per-capita growth is due to embodied technical change.
As pointed out by Hercowitz (1998), Gordon’s series have been good news for the
Solowian view.
Replacement Echoes. Solow et al. (1966) examine the polar case, where factor
substitution is not allowed neither ex-ante nor ex-post. Under Leontief technology,
Y (v,t) = Y (v) = I(v) = eγv L(v), for all t ≥ v. One unit of vintage capital v produces
one unit of output once combined with e−γv units of labor. Technical progress is
embodied and takes the form of a decreasing labor requirement. For the same reasons
as in Johansen, capital goods are scrapped at ﬁnite time.
Under constant saving rate, and some technical assumptions, Solow et al. show con-
vergence to a unique balanced growth path, delivering the same qualitative asymptotic
behavior as the neoclassical growth model. This was quite disappointing, since under
ﬁnite lifetime one would have expected investment burst from time to time, giving
rise to the so-called replacement echoes.
Let normalize the labor supply to unity. From labor market clearing,
R t
t−T(t) L(v) dv =
1. Under constant lifetime, time diﬀerentiation of the equilibrium condition yields
L(t) = L(t − T), implying that investment is mainly driven by replacement activ-
ities. When obsolete capital is destroyed, new investments are needed to replace
the scrapped machines, creating enough jobs to clear the labor market. As a direct
consequence, job creation and investment have a periodic behavior, implying that
investment cycles are reproduced again and again in the future.
Solow et al did not ﬁnd echoes because of the constant saving rate assumption, which
4completely decouples investment from replacement. In an optimal growth model
with linear utility and the same technological assumptions, Boucekkine, Germain and
Licandro (1997) show (ﬁnite time) convergence to a constant lifetime, letting replace-
ment echoes operate and generate everlasting ﬂuctuations in investment, output and
consumption. Under strictly concave preferences, ﬂuctuations do arise in the short-
run but get dampened in the long-run by consumption smoothing (see Boucekkine et
al., 1998). Therefore, the short-run dynamics of vintage capital models strikingly dif-
fer from the neoclassical growth model, provided capital and labor are to some extent
complementary, consistently with the observed dynamics of investment both at the
plant level (Doms and Dunne, 1998) and the aggregate level (Cooper, Haltiwanger
and Power, 1999). Non-monotonic behavior has also been shown by Benhabib and
Rustichini for vintage models with non-geometric depreciation.
Vintage human capital. The vintage capital growth literature typically considers
labor as a homogenous good. However, like physical capital is heterogenous, so is the
labor force. The concept of vintage human capital has been explicitly used in the 90s
to treat some speciﬁc issues related to technology diﬀusion, inequality and economic
demography.
Technology diffusion. In Chari and Hopenhayn (1991) and Parente (1994),
individuals face the dilemma of whether to stick to an established technology or to
adopt a new and better one. The trade-oﬀ is the following: adopting allows the
use of an advanced technology at the cost of loosing expertise, the speciﬁc human
capital accumulated on the currently used technique. Chari and Hopenhayn model it
in a two-period overlapping generations model where diﬀerent vintage technologies,
operated by skilled and unskilled workers, coexist. Old workers are experts in the
speciﬁc vintage technology they have run when young. The degree of complementary
between skilled and unskilled labor aﬀects negatively the velocity of technological
diﬀusion, since young individual have large incentives in investing in old technologies
when their unskilled labor endowment is highly complementary to the skilled labor of
the old.
Inequality. Jovanovic (1998) argues that vintage capital models are particularly
well suited to explain income disparities across individuals and across countries. As in
Johansen, diﬀerent vintages coexist even though new machines are more productive.
Under the assumption that machines’ quality and labor’s skill are complementary, the
best machines are operated by the best skilled individuals, exacerbating inequality.
5Human capital accumulation drives growth by promoting the investment in new vin-
tage capital technologies. The best skilled workers are immediately assigned to the
frontier technology, the second bests go to the machines just below the frontier, and
so on. Even if it goes at odds with Chari and Hopenhayn, where adoption costs in-
duce a much slower switching of technologies, frictionless reassignment has the virtue,
consistent with cross country evidence, of implying persistent inequality in contrast
to Parente (1994) which bears leapfrogging.
Demographics. One likely channel through which demographics aﬀect growth is
the size, quality and composition of the work force. In this perspective, generations
of workers can be understood as being vintages of human capital. In a continuous
time overlapping generations framework, Boucekkine, de la Croix and Licandro (2002)
model the vintage speciﬁcity of human capital from schooling decisions. Individuals
optimally decide how many years to spend at school as well as their retirement age;
life expectancy has a positive eﬀect on both, because of its beneﬁcial impact on the
return of education. In such a framework, the vintage speciﬁcity of human capital
does not depend on technological vintages as in Chari and Hopenhayn (1991) but on
cohort speciﬁc demographic characteristics, including education.
The observed relation between demographic variables, such as mortality, fertility and
cohort sizes, and growth is anything but linear. Since a key element is between-
generation diﬀerences in human capital, these nonlinearities may be modeled by the
mean of a vintage structure of population. Boucekkine et al. generate nonlinear rela-
tionships between economic growth and both population growth and life expectancy.
A longer life, for example, has several conﬂicting eﬀects. On one hand, it raises the
incentives to educate and reduces the depreciation rate of aggregate human capital.
But on the other, an older population, who did their schooling a long time ago, is
harmful for economic growth.
References
Benhabib J. and A. Rustichini (1991), “Vintage Capital, Investment, and Growth.”
Journal of Economic Theory 55, 323–339.
Boucekkine R., D. de la Croix and O. Licandro (2002), “Vintage human capital,
demographic trends and growth.” Journal of Economic Theory 104, 340–375.
6Boucekkine R., M. Germain and O. Licandro (1997) “Replacement echoes in the
vintage capital growth model.” Journal of Economic Theory 74, 333–348.
Boucekkine R., M. Germain, O. Licandro and A. Magnus (1998), “Creative destruc-
tion, investment volatility and the average age of capital.” Journal of Economic
Growth 3, 361–384.
Chari V.V. and H. Hopenhayn (1991), “Vintage human capital, growth, and the
diﬀusion of new technology.” Journal of Political Economy 99, 1142–1165.
Cooper R., J. Haltiwanger and L. Power (1999), “Machine replacement and the busi-
ness cycle: Lumps and bumps.” American Economic Review 84, 921–946.
Denison E. (1964), “The unimportance of the embodied question.” American Eco-
nomic Review - Papers and Proceedings 54, 90–94.
Doms M. and T. Dunne (1998), “Capital adjustment patterns in manufacturing
plants.” Review of Economic Dynamics 1: 409–429.
Gordon R. (1990), The measurement of durable goods prices. University of Chicago
Press.
Greenwood J., Z. Hercowitz and P. Krusell (1997), “Long-run implications of invest-
ment speciﬁc technological change.” American Economic Review 87, 342–362.
Hercowitz Z. (1998), “The embodiment controversy: A review essay.” Journal of
Monetary Economics 41, 217–224.
Johansen L. (1959), “Substitution Versus ﬁxed production coeﬃcients in the theory
of economic growth.” Econometrica 27, 157–176.
Jovanovic B. (1998), “Vintage capital and inequality.” Review of Economic Dynamics
1, 497–530.
Parente S. (1994), “Technology adoption, learning-by-doing, and economic growth.”
Journal of Economic Theory 63, 346–369.
Solow R. (1960), “Investment and technological progress.” In K. Arrow, S. Karlin and
P. Suppes (eds.), Mathematical Methods in Social Sciences 1959, 89–104. Stanford
University Press.
7Solow R., J. Tobin, C. Von Weizsacker and M. Yaari (1966), “Neoclassical growth
with ﬁxed factor proportions.” Review of Economic Studies 33, 79–115.
Whelan, K. (2002), “Computers, obsolescence and productivity.” The Review of Eco-
nomics and Statistics 84, 445–461.
8Département des Sciences Économiques
de l'Université catholique de Louvain
Institut de Recherches Économiques et Sociales
Place Montesquieu, 3
1348 Louvain-la-Neuve, Belgique
 ISSN 1379-244X         D/2006/3082/014

8Département des Sciences Économiques de l'Université catholique de Louvain Institut de Recherches Économiques et Sociales Place Montesquieu,

Capital adjustment patterns in manufacturing plants.”

Computers, obsolescence and productivity.”

Creative destruction, investment volatility and the average age of capital.”

Investment and technological progress.” In

Long-run implications of investment speciﬁc technological change.”

Machine replacement and the business cycle: Lumps and bumps.”

Neoclassical growth with ﬁxed factor proportions.”

Replacement echoes in the vintage capital growth model.”

Substitution Versus ﬁxed production coeﬃcients in the theory of economic growth.”

Technology adoption, learning-by-doing, and economic growth.”

The embodiment controversy: A review essay.”

The measurement of durable goods prices.

The unimportance of the embodied question.” American Economic Review - Papers and

Vintage Capital

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