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The need to introduce heterogeneity in the capital-accumulation process and to differentiate capital goods by their vintage or productivity have been widely recognized since the seminal work of Solow [37]. Together with embodied technological progress, the most crucial aspect of the vintage capital models is that capital goods of later date are more productive, or make products of higher quality (see [28], [38], [2] and [7]). Accordingly, recent studies (see among others, [40], [16], [1], [17]) take into account the fact that firms not only have to decide on the volume of the capital goods but also on the optimal age distribution of them, and have been centered around the questions posed by Chari and Hopenhayn [12]: Why are new technologies often adopted so slowly? Why do people of-ten invest in old technologies even when apparantly superior technologies are available? How are decisions to adopt new technologies affected by the prospect that even better technologies will arrive in the future?

Vintage capital models are able to generate different properties and dynamics from the classical capital accumulation models. The so-called “vintage effect”, i.e. the productivity differential between successive vintages of capital due to embodied technological progress, plays a crucial role on the firm’s optimal plans. Such a productivity differential creates an advantage in investments in younger machines. Younger machines are not only endowed with a superior technology but have also a longer lifetime than the older ones (see [5]). However, they have also the disadvantage that the older machines are cheaper and the costs of depreciation and discounting are less. With the presence of these latter effects, [16] provides an explanation for why new technologies are often adopted on a large scale only after a long period of time. Taking into account the vintage effect, [16] analyzes in what way a perfectly competitive firm adjusts current investments to the predictions of technological progress. As current investments do not affect the profitability of investments in future technologies in a perfectly competitive market, predictions of higher technological progress in the future do not influence the current investments. However, considering market power, [17] shows that a ”negative anticipation effect” occurs. Since current investments increase output which decreases the price, this creates a negative effect on the profitability of future investments, so that the anticipated technology shocks will be preceded by declines of investment. This will be followed by a period of higher growth where new capital goods can be purchased without reducing the output price too much.

A common assumption in these studies and also in most of the macroeconomic literature is that the depreciation rate is either exogenously fixed or acts as a residual (depreciation-in-use hypothesis) independent of maintenance activities. However, empirical (see among others, [23], [32], [30], [25]) and theoretical (see [20], [33], [9]) findings state that depreciation rate is neither constant, as it increases with the ageing of the capital stock, nor a residual variable as it can be controlled by the economic agents by choosing appropriate levels of maintenance expenditures. Indeed, many firms have in mind the maintenance implications of their adoption decisions. Even a firm can disregard the adoption of a new technology if it anticipates a costly pace of maintenance costs. As argued by McGrattan and Schmitz [30], maintenance expenditures are too important to be neglected even at the aggregate level and hence, become inevitable when analyzing how the firms set their optimal plans. The endogenous nature of depreciation rate depending on maintenance expenditure also highlights the trade-offs between new investment and higher maintenance expenditures and induces the following questions to emerge in order to reach a better understanding of the firm’s capital accumulation process:

i) Under what conditions do maintenance and investment appear as substitutes or complements?
ii) What is the optimal allocation of investment and maintenance across vintages?
iii) How do maintenance and investment decisions shift in response to an exogenous change in the rate of technological progress?
iv) Can there be a negative anticipation effect even under non-monopolistic settings?

The economic literature devoted to the role of maintenance in the economy is mostly composed of empirical or computational studies concerned with the cyclical properties of maintenance and its implications for the business cycle (see among others [29] and [14]. Among very few theoretical contributions, [6] studies the optimal allocation of labor resources to production, technology adoption or capital maintenance in a one-hoss-shay vintage capital model and points out that though capital maintenance deepens the technological gap by diverting labor resources from adoption, it generally increases the long run output level at equilibrium. In a very recent study, how technological progress both embodied and disembodied affects the life-time of capital have been analyzed by [4]. Apart from these studies concerned with the economic performances at the aggregate level, there is no accompanying theoretical contribution in a vintage capital framework at the firm level. [26] and [9] investigate the demand for maintenance services in some typical firms’ investment problems with a deliberate microeconomic approach. However, both studies do not take into consideration the ageing of the capital stock and can not attempt to answer the questions (ii) and (iv).

This paper provides the needed analysis of the firm’s capital accumulation process taking into account that depreciation is endogenous and in particular associated with vintage specific maintenance expenditure. We present precise answers to the questions (i)-(iv) with a detailed analysis of the substitutability between investment and maintenance services. We provide an intrinsic definition of complementarity vesus substitutability and prove that maintenance is a local substitute for investment as soon as the marginal cost of maintenance is strictly increasing. We show that maintenance and investment in new capital goods appear as complements with respect to the changes in productivity, cost of maintenance, fixed cost of operation and the efficiency of maintenance services. We also find that investment in new capital goods and maintenance services are substitutes in the traditional sense: when the price of new machines changes, the demands for capital goods and for maintenance respond in the opposite direction. We analyze the effects of technological progress on the firm’s optimal plans and prove that a negative anticipation effect can occur even without any market imperfections. Our set up allows for an extension where investments in the old vintages are possible. We show that investment in old machines appears as a substitute of both investments in new machines and maintenance services.

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