The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Vintage Capital

  • Raouf Boucekkine
  • David de la Croix
  • Omar Licandro
Reference work entry


This article reviews the early vintage capital literature of the 1960s, and identifies the factors behind the revival of topic from the 1990s. The fundamental properties of the seminal vintage capital growth models are disentangled, and the origins of the associated controversy on the importance of embodied technical progress are evoked. The recent revival of this literature is analysed with special emphasis on the rising support for the Solowian view of investment following Gordon’s 1990 fundamental work on the price of durable goods, and the emergence of a new vintage human capital literature devoted to some fundamental economic growth issues.


Capital–labour substitution Cobb–Douglas technology Cohort size Consumption smoothing Demographic transition Demography Denison, E. Depreciation Diffusion of technology Embodiment hypothesis Fertility Human capital Inequality (explanations) Information technology revolution Innovation Investment cycles Leontief technology Life expectancy Neoclassical growth theory Neoclassical production functions Obsolescence Overlapping generations models Phelps, E. Population growth Productivity slowdown Quasi-rent Replacement echoes Solow, R. Technical progress Transition dynamics Vintage capital Vintage human capital 
In neoclassical growth theory capital is assumed to be homogeneous and technical progress disembodied, meaning that all capital units benefit equally from any technological improvement. The disembodied nature of technical progress looks 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 affect 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. It actually inspired an important stream of the growth literature of the 1950s and 1960s, giving birth to 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 depreciation 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
$$ Y\left(v,\kern0.5em t\right)=F\left(I(v),L\left(v,t\right),{e}^{\upgamma v}\right), $$
where Y(v, t) is the output of vintage v at time tv and L(v, t) is the amount of labour assigned to this vintage. Parameter γ > 0 designates the rate of technical progress, which is said to be embodied since it benefits only vintage v. F(.) has the properties of a neoclassical production function. Vintages produce the same final good
$$ Y(t)={\int}_{t-T(t)}^tY\left(v,\kern0.5em t\right)\kern0.5em dv, $$

where Y(t) is total production and T(t) is the lifetime of the oldest operative vintage.

Besides realism, vintage capital models were initially thought to be able to generate quite different long-run properties and short-term dynamics from neoclassical growth models. Because the productivity gap between new and old vintages is increasing over time, the latter need not be operated for ever, and, contrary to the neoclassical growth theory, the lifetime of capital goods might well be finite (Johansen 1959). Such a property was thought to involve non-monotonic transition dynamics governed by the replacement of scrapped goods, known as the replacement echoes principle, which again sharply departs from neoclassical growth models.

On more general ground, vintage capital models were at the heart of the embodiment controversy, which opposed Solow to some leading growth theorists and empiricists, among them Phelps (1962) and Denison (1964). While the former argued that accounting for the fraction of technological progress which is exclusively conveyed by capital accumulation (that is, embodied technical progress) is important to accounting for growth, Phelps argued that the decomposition of technical progress is irrelevant in the long run. Recent studies notably by Gordon (1990) have resuscitated this controversy, as we shall see. Before developing all these themes, it should be noted that, whereas early vintage capital theory primarily focused on physical capital accumulation, recent contributions have taken the same view of human capital accumulation (see Chari and Hopenhayn 1991). Vintage human capital is generated either by successive vintages of technologies which induce specific skills or by demographic conditions. Such contributions have brought out a new and quite appealing understanding of the mechanisms behind technology diffusion and demographic transitions, for example. We briefly review them also.

The Lifetime of Capital

In Johansen (1959), technology is ‘putty-clay’, meaning that capital–labour substitution is permitted ex ante but not once capital is installed. Technological progress is assumed to be labour-saving. Because factor proportions are fixed ex post,
$$ Y\left(v,\kern0.5em t\right)=F\left(I(v),\kern0.5em {e}^{\upgamma v}L\left(v,\kern0.5em t\right)\right)=g\left(\lambda (v)\right)\kern0.5em I(v), $$

where the labour–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 vintage v at date t are proportional to g(λ(v)) − λ(v) eγvw(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.

Replacement Echoes

If capital lifetime is finite, there might be a room for replacement echoes, as mentioned above. Solow et al. (1966) examine this question in the simpler case of a Leontief technology, when factor substitution is not allowed either ex ante or ex post. In such a case, Y(v, t) = Y(v) = I(v) = eγvL(v), for all tv. One unit of vintage capital v produces one unit of output once combined with eγv units of labour.

Technical progress is embodied and takes the form of a decreasing labour requirement. For the same reasons as in Johansen, capital goods are scrapped in finite time. Using in addition a constant saving rate, and some technical assumptions, Solow et al. show convergence to a unique balanced growth path, delivering the same qualitative asymptotic behaviour as the neoclassical growth model. This was quite disappointing, since under finite lifetime one would have expected an investment burst from time to time, giving rise to replacement echoes.

Let us normalize the labour supply to unity. From labour market clearing, \( {\int}_{t-T(t)}^tL(v)\kern0.5em dv=1 \). Under constant lifetime, time differentiation of the equilibrium condition yields L(t) = L(tT), implying that investment is mainly driven by replacement activities. When obsolete capital is destroyed, new investments are needed to replace the scrapped machines, creating enough jobs to clear the labour market. As a direct consequence, job creation and investment have a periodic behaviour, implying that investment cycles are reproduced again and again in the future.

Solow et al. (1966) did not find echoes because of the constant saving rate assumption, which completely decouples investment from replacement. In an optimal growth model with linear utility and the same technological assumptions, Boucekkine et al. (1997) show (finite time) convergence to a constant lifetime, letting replacement 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 differ strikingly from the neoclassical growth model, provided capital and labour are to some extent complementary, consistently with the observed dynamics of investment at both the plant level (Doms and Dunne 1998) and the aggregate level (Cooper et al. 1999). Non-monotonic behaviour has also been shown by Benhabib and Rustichini (1991) for vintage models with non-geometric depreciation.

The Embodiment Hypothesis

A crucial property of vintage capital models is the embodied nature of technological progress: the incorporation of innovations into the production process cannot be achieved without the acquisition of the new vintages which are their exclusive material support. According to Solow (1960), embodiment can have crucial implications for growth accounting. To make the point, he considers a Cobb–Douglas vintage technology
$$ Y\left(v,t\right)={\left[{\mathrm{e}}^{\gamma v}\kern0.5em I(v)\right]}^{1-\alpha}\kern0.5em L{\left(v,\kern0.5em t\right)}^{\alpha }, $$

and the capital–labour 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 arbitrarily small amounts of labour.

A striking outcome of Solow’s model is its aggregation properties. Denote by L(t) the total labour supply, and define quality adjusted capital as
$$ K(t)={\int}_{-\infty}^t{\mathrm{e}}^{\gamma v}I(v)\kern0.5em dv. $$
Since marginal labour productivity equalizes across vintages, aggregate output becomes
$$ Y(t)=K{(t)}^{1-\alpha }L{(t)}^{\alpha }. $$
Aggregate vintage technology in Solow (1960) degenerates into a neoclassical production function. However, by differentiating (1), the motion law for capital is slightly different
$$ {K}^{\prime }(t)={\mathrm{e}}^{\gamma t}I(t), $$
reflecting embodied technical change. Since eγt measures the relative price of investment goods at equilibrium, the value of capital is by definition A(t) = eγtK(t), and evolves following
$$ {A}^{\prime }(t)=I(t)-\upgamma A(t). $$

Technological progress operates as a steady improvement in equipment quality, which in turn implies obsolescence of the previously installed capital. In Solow, obsolescence does not show up through finite time scrapping but through labour reallocation reflecting a declining value of capital.

This important point has been at the heart of recent literature on the productivity slowdown and the information technology revolution (see Whelan 2002). Indeed, the potential implications for growth of embodied technical progress were tremendously controversial in the 1960s. In a famous statement, Denison (1964) claimed ‘the embodied 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 channel. Greenwood et al. (1997), by using Gordon’s (1990) estimates of the relative price of equipment, quantitatively evaluate the two-sector Solow model, claiming that around 60 per cent 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.

Vintage Human Capital

The vintage capital growth literature typically considers labour as a homogenous good. However, just as physical capital is heterogenous, so too is the labour force. The concept of vintage human capital was explicitly used in the 1990s to treat some specific issues related to technology diffusion, inequality and economic demography.

In a world with a continuous pace of innovations, a representative individual faces the typical question of whether to stick to an established technology or to move to a new and better one. The trade-off is the following: switching to the new technique would allow him to employ a more advanced technology but he would lose the expertise, the specific human capital, accumulated on the old technique. In Chari and Hopenhayn (1991) and Parente (1994), individuals face exactly this dilemma. In such frameworks the generated vintage human capital distributions essentially mimic the vintage distribution of technologies, the time sequence of innovations being generally exogenously given. Chari and Hopenhayn (1991) consider 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 labour affects negatively the velocity of technological diffusion, since young individuals have strong incentives to invest in old technologies when their unskilled labour endowment is highly complementary to the skilled labour of the old.

Jovanovic (1998) argues that vintage capital models are particularly well suited to explain income disparities across individuals and across countries. The main mechanism behind them is the following. Under the assumption that machines’ quality and labour’s skill are complementary, the best machines are assigned to the best-skilled individuals, exacerbating inequality. If reassignment is frictionless, then the best-skilled workers are immediately assigned to the frontier technology, the second-best go to the machines just below the frontier, and so on. Even though it is at odds with Chari and Hopenhayn, where adoption costs induce 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.

On the theoretical side, Jovanovic makes an important contribution to the vintage capital literature to the extent that he addresses the hard problem of combining vintage physical capital and vintage human capital in a framework where the vintage distributions of both assets are endogenous. Jovanovic uses an assignment model à la Sattinger (1975) to solve this difficult problem. Firms combine machines and workers in fixed proportions, say one machine for one worker, at every instant. Because labour resources are fixed, the latter fixed-proportions assumption implies that old machines become unprofitable at a finite time, as in Johansen. Vintage human capital comes from human capital accumulation à la Lucas (1988): the growth of the stock of human capital determines the maximal quality of human capital available: if the worker has human capital, h, and works a fraction of time u (in production), then her skill is given by s = u h. The typical assignment problem of a firm having acquired capital of a given vintage is to find the optimal vintage human capital or skill of the associated worker (via profit maximization), which makes it possible to achieve the pairing of skills and machines on the basis of the persistent inequality mechanism outlined above.


One likely channel through which demographics affect growth is the size, quality and composition of the workforce. From this perspective, generations of workers can be understood as being vintages of human capital. In a continuous-time overlapping generations framework, Boucekkine et al. (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 to education. In such a framework, the vintage specificity of human capital depends, not 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 modelled by the mean of a vintage structure of population. Boucekkine et al. (1998) generate nonlinear relationships between economic growth and both population growth and life expectancy. A longer life, for example, has several conflicting effects. On the one hand it increases the incentives to acquire education and reduces the depreciation rate of aggregate human capital. But on the other, an older population, which finished its schooling a long time ago, is harmful for economic growth.


After a relatively long stagnation, the vintage capital literature, which was a fundamental growth area in the 1960s, has been experiencing a revival since the early 1990s. This revival is due to several factors, among them the rising support for the Solowian view of investment following Gordon’s fundamental work on the price of durable goods, the emergence of a new vintage capital growth theory led by Benhabib and Rustichini (1991) relying on a novel and appropriate mathematical set-up, and notably the increasingly common view that some fundamental economic growth issues (like technology diffusion, for example) do require the vintage structure to be better appraised. Of course, many tasks within this new literature remain to be addressed. In particular, much work is needed to bring the vintage models closer to the data. The work of Gilchrist and Williams (2000) is fundamental is this respect.

See Also


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Copyright information

© Macmillan Publishers Ltd. 2018

Authors and Affiliations

  • Raouf Boucekkine
    • 1
  • David de la Croix
    • 1
  • Omar Licandro
    • 1
  1. 1.