Obsolescence Vs modernization in a Schumpeterian vintage capital model by Raouf Boucekkine* Fernando del Río** Omar Licandro*** DOCUMENTO DE TRABAJO 2000-27

December, 2000

* Université Catholique de Louvain and IRES. ** Universidade de Santiago de Compostela. *** FEDEA.

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Obsolescence Vs modernization in a Schumpeterian vintage capital model∗

Raouf Boucekkine† Fernando del Río‡ Omar Licandro§

December, 2000

Abstract

In this paper, we build up a general equilibrium model explicitly incorporating Schumpeterian growth `a la Aghion and Howitt (1992) and a vintage capital structure in line with Solow (1960). Technological progress is embodied. We show that the investment rate is a fundamental determinant of the profitability of R&D in contrast to the R&D based growth models with disembodied technical progress. We characterize the balanced growth paths and point at the possible existence of multiple equilibria due to the strategic complementarity between investment and R&D activities. More importantly, the embodiment hypothesis is shown to give rise to a precise modernization mechanism through investment and the average age of capital. The modernization efects of investment may well balance the typical obsolescence costs inherent to embodiment, a result that should be of interest in the current debate on the viability of the current information technology boom as a long run growth regime.

Keywords: Vintage capital, R&D, Creative destruction, Embodiment, Obso lescence, Modernization.

Journal of Economic Literature: E22, E32, O40.

∗We would like to thank Bruno Amable, David de la Croix, Victor Ríos-Rull and the participants at the II Louvain Symposium on macroeconomic dynamics for useful comments. Any remaining error is ours.

†Corresponding author. Université catholique de Louvain and IRES, Place Montesquieu, 3, B-1348 Louvain-la-Neuve (Belgique). E-mail: boucekkine@ires.ucl.ac.be.

‡Universidade de Santiago de Compostela. E-mail: aedelrio@usc.es

§FEDEA. E-mail: licandro@fedea.es.

1 Introduction

Since 1995, the US economy has been experiencing a spectacular recovery with huge figures for both the growth rates of GDP and productivity. The sector of information technologies (IT hereafter) has been repeatedly invoked to explain this resurgence of growth (see among others Gordon, 1999, Jorgenson and Stiroh, 2000, and Oliner and Sichel, 2000). In particular, the observed massive investment in these technologies has been pointed out by many authors and practitioners as the main determinant of the current state of the US economy. For example, Alan Greenspan has recently claimed that: “...The newer technologies and foreshortened lead times apparently have made capital investment distinctly more profitable, enabling firms to substitute capital for labor and other inputs far more productively than they could have a decade or two ago. Capital, as economists like to say, has deepened significantly since 1995”.¹

One common interpretation of the crucial role of investment in the ongoing technological revolution puts forward the concept of embodied technological change. As the technological progress associated with IT is mainly embodied, investment is the unique vehicle of progress. Massive accumulation of capital goods is stimulated by the investment-specific nature of technological progress which favors the observed acceleration in the relative price of equipment. The latter phenomenon is even more striking for IT equipment as emphasize Jorgenson and Stiroh (2000): The price of computer investment fell around 17% per year from 1990 to 1996. Does this mean that we are entering a new growth regime driven by IT investment, with higher growth rates for GDP and productivity? There is an ongoing debate on this question.² One argument against this optimistic view is the obsolescence costs inherent to embodied technological progress. An elementary illustration of the negative efects of those costs on (long-run) growth can be found in Aghion and Howitt (1998), chapter 3. A more recent theoretical argument is due to Boucekkine, del Río and Licandro (1999b). Using a two-sector model with learning-by-doing driving technological progress in both sectors, they show that a reassignment of learning eficiency from the consumption goods sector to the capital goods sector induces a permanent slowdown in the growth rates of GDP and productivity because of the embodied characteristic of technological progress in the latter sector with the associated obsolescence costs.

However, because learning-by-doing is a rough simplification of the way technological advances are taking place, a more comprehensive modeling of the latter is certainly needed. This paper presents a Schumpeterian research-based multi-sector growth model with embodied technological progress. With respect to Aghion and Howitt (1998), chapter 3, we use the same modeling strategy for the research sector but the embodiment characteristic of technological progress, through a vintage capital structure in the final goods sector, allows us to analyze in a much less ad-hoc manner the efects of obsolescence costs on growth within the same Schumpeterian set-up. In addition to this contribution, this paper puts forward a key role of investment when technological progress is embodied, its modernization efects. The fundamental variable involved in the modernization mechanism is the average age of capital. In our model, an increase in the investment rate stimulates growth and lowers the average age of capital. Indeed the same properties would be obtained if technological progress were disembodied, as in Aghion and Howitt (1993), chapter 3. However in our model, because of the embodiment hypothesis, an additional efect is shown to take place: The decrease in the average age of capital implies a further rise in the investment rate with the subsequent upward shift in the growth rate and downward shift in the average age of capital. And so forth. This modernization mechanism may well balance the obsolescence costs inherent to embodiment. Consequently this model delivers a much less pessimistic view of the current IT revolution.

¹A. Greenspan, Chicago, May 6, 1999.

²The latest performances of the US economy have stimulated this debate even more.

Our model can be seen as a synthesis between the vintage capital literature and the Schumpeterian growth literature. Vintage capital structures turn out to be most useful to capture some of the most salient characteristics of investment activity and job allocation in real economies. Because the vintage structure is the natural one to treat the investment-replacement-adoption decisions when technological progress is embodied in the new capital goods, it has been intensively used in the recent years to study the foundations of lumpy investment behaviour and its implications on business fluctuations (see Cooley et al., 1997, and Boucekkine et al.,1999a, for more details). On the other hand, the impressive magnitude and the qualitative characteristics of job creation and destruction at the plant level have been abundantly documented in the recent literature (see also Dunne et al., 1989, and Davis et al., 1996), and all these contributions can be advocated to motivate a vintage approach. Finally, there is a substantial evidence of investment irreversibility at the plant level (see in particular Caballero et al., 1995). The new plants typically incorporate the new capital inputs, which in turn embody the most recent technological advances, while the older plants only use the older generations of capital goods and the less eficient technologies. This induces an economic obsolescence mechanism and a continuous creative destruction process: While new plants are created with the up-to-date technology and capital inputs, the oldest are meanwhile eventually destroyed. It therefore clearly appears that vintage capital models are a natural setup for the Schumpeterian creative destruction to take place.

This Schumpeterian concept of creative destruction is also at the basis of one of the most influential research-based endogenous growth theories of the recent years, namely Aghion and Howitt’s model (1992). In this model, economic growth is generated thanks to a random sequence of innovations improving the quality ladders of the goods. The innovations result from intended R&D activities with random returns. Creative destruction arises as the better quality new goods cause the obsolescence of the older ones. Caballero and Jafé (1993) and Barro and Sala-i-Martin (1995), chapter 7, have provided multi-sectoral extensions of Aghion and Howitt’s setting. More recently, Aghion and Howitt (1998), chapter 3, and Howitt and Aghion (1998) have amended Caballero and Jafé’s model so as to include capital accumulation. More precisely, Aghion and Howitt have combined within the same framework the basic characteristics of Solow’s neoclassical growth model (1956) and those of their creative destruction model. As one can infer from reading the two contributions mentioned just above, the incorporation of the creative destruction story into the neoclassical growth model requires a substantial adaptation. We argue in this paper that a much simpler and more meaningful way to introduce capital accumulation into Aghion and Howitt’s creative destruction model consists in combining the latter with another Solow’s fundamental contribution, namely his vintage capital model (1960). Since vintage capital models are consistent with creative destruction, combining them with Schumpeterian R&D specifications seem natural and even desirable.

There are some expected gains from this combination in terms of the explanatory power of the theory. In contrast to the basic creative destruction model, ours will reproduce the fundamental stylized facts linked to embodiment. Indeed, we will show that the rate of decline of the relative price of capital is negatively correlated with the growth rate of the economy along the balanced growth path. The predictions of our theory regarding the evolution of the investment to output ratio and job reallocation across plants are also completely consistent with the empirical studies mentioned above. Less immediate gains from our theory concern the potential multiplicity of long run equilibria. It is well known since Cooper and John (1988) that the presence of strategic complementarities favors the emergence of multiple equilibria. Our model exhibits a strategic complementarity between investment and R&D activities. Indeed, one of its most salient characteristics is that the marginal return to R&D depends positively on the investment rate. This strategic complementarity is a potential source of multiple equilibria in our setup. The occurrence of multiplicity in endogenous growth models has been very often stressed in the recent literature (see the comprehensive survey of Evans et al., 1998). Our model shows that the embodiment hypothesis (through our vintage structure) is a plausible source of multiplicity in a Schumpeterian creative destruction setup.

As mentioned above, one major characteristic of our model is that investment is involved in a very precise modernization mechanism yielding higher growth rates and lower average ages of the capital stock. The crucial role of investment in the growth process has been recently outlined in several empirical and theoretical contributions, often in connection with embodied technical change. For a sample of seven OECD countries, Wolf (1991) has found that catch-up in total factor productivity depends on capital accumulation, and that embodiment plays a central role in this respect. De-Long and Summers (1991) evidence according to which the countries with the higher growth rates are those who experienced higher investment rates in equipment and faster declines in the relative price of capita, is also consistent with our view of the role of investment. Recently, McGrattan (1998) revisited the arguments put forward by Jones (1995) to question the tight correlation between investment and growth rates as implied by the AK models. In particular, she showed that if longer time series and more countries are considered in the data, this tight correlation is definitely preserved.

Because of the strength of the modernization efects of investment in our model, our conclusions as to the eficiency of capital subsidy are diferent from those of Aghion and Howitt (1998), chapter 3. According to these authors, the magnitude of the (positive) efect of capital subsidy on growth is lower as far as obsolescence costs are involved. In our model, the strong modernization efects of investment may well rule out this property. On the other hand, our analysis of the latter efects can be related to the well known “embodiment controversy”. The view advocated by Phelps (1962) according to which embodiment and the associated modernization efects of investment are only important in the short run, is also shown to not hold in our research-based endogenous growth framework in the last section of this paper.

This paper is organized as follows. In the next section, we present our model in details and stress its main properties. Section 3 derives the balanced growth paths of the model and its comparative statics. Section 4 is devoted to the analysis of the eficiency of capital and research subsidies in connection with the study previously carried out by Aghion and Howitt (1998), chapter 3, and Howitt and Aghion (1998). Section 5 concludes.

2 The model

As announced in the introduction, our model can be seen as a mixture of Solow’s vintage capital model (1960) and the Schumpeterian set-up `a la Aghion-Howitt (1992). Solow’s production function with vintage capital is introduced in the final goods sector, and the specification of the creative destruction process in the sector of intermediate capital goods is in line with Aghion and Howitt’s story. Here are the salient features of our specifications.

Vintage capital in the final goods sector

The economy produces a unique final good which can be either consumed, invested or devoted to research activities. As usual, we assume that the final goods sector is competitive. To introduce our vintage structure, we also assume that the final good is produced by a continuum of plants which difer by their age. More precisely, we hypothesize that the production at time t of a plant created at date z (therefore of age t z) is given by:

\[y (z, t) = B l (z, t) ^ {\alpha} \int_ {0} ^ {M} \left(q ^ {\kappa_ {j} (z) \tau} k _ {j} (z, z) e ^ {- \delta (t - z)}\right) ^ {1 - \alpha} d j,\tag{1}\]

where is the amount of labor used at time t to operate the plant constructed at time z, being B and two positive constants with . All the plants are assumed to use a continuum of capital goods, . Accordingly, is the amount of the j-th capital good used at time z by the plant created at time z. Note that from (1), all capital goods depreciate over time at the same rate δ. represents the quality grad of the j-th capital good used by the plant built at being a positive constant greater than 1, and τ another positive constant. represents the number of quality improvements of the j-th capital good achieved up to date z. For convenience, we assume that quality grad of the j-th capital good is at thanks to a single past innovation.³ A second innovation increases the quality grad to , a third innovation to , and so forth. Since denotes the number of innovations achieved up to period , the maximum quality index available at the j-th capital good is . In the specification of the production function of the plant built at period z just above, we assume that this plant only uses the best quality of any capital good available at z, a property which will be shown to hold at equilibrium.

Summing up the outputs of all the (active) plants at time we get the aggregate production, :

\[Y (t) = \int_ {- \infty} ^ {t} B l (z, t) ^ {\alpha} \int_ {0} ^ {M} \left(q ^ {\kappa_ {j} (z) \tau} k _ {j} (z, z) e ^ {- \delta (t - z)}\right) ^ {1 - \alpha} d j d z.\tag{2}\]

The obtained vintage capital production function is similar to the specification discussed in details by Solow (1960). In particular, the characteristics of the adopted vintage technology imply that it is never optimal to close a plant whatever its age

We now turn to the optimal behaviour of the firms. In our model, the representative firm has to choose the optimal amount of each type of capital in order to build up a new plant, and the optimal allocation of labor to the plants. This is done by maximizing discounted profits, given prices and the technological constraint (1). We assume that capital accumulation is subsidized at a rate , so that the discounting factor of the firms is . The resulting first order conditions for an interior maximizer to exist are:

\[(1 - \alpha) B q ^ {(1 - \alpha) \kappa_ {j} (t) \tau} k _ {j} (t, t) ^ {- \alpha} \int_ {t} ^ {\infty} e ^ {- \int_ {t} ^ {z} (r (s) + \delta (1 - \alpha) - \beta_ {k}) d s} l (t, z) ^ {\alpha} d z = p _ {\kappa_ {j}} (t),\tag{3}\]

\[\alpha B l (z, t) ^ {\alpha - 1} \int_ {0} ^ {M} \left(q ^ {\kappa_ {j} (z) \tau} k _ {j} (z, z) e ^ {- \delta (t - z)}\right) ^ {1 - \alpha} d j = w (t),\tag{4}\]

where is the interest rate at date is the wage at and is the price of the j-th capital good at t. We take the final good as the numeraire. Equation (3)

q^τ

nl→0 ∂y ∂l ⁼ ∞ al =∞

³This assumption is an innocuous normalization. Our implicit assumption that all the capital goods “start” with the same quality grad is also innocuous, and could be perfectly relaxed.

⁴Indeed, the specified production function is such that lim_{l 0}

determines the demand for the j-th capital good, it simply states that the discounted marginal return to the j-th capital good is equal to its price. Equation (4) establishes the way labor has to be optimally assigned to the plant built up at time z: The marginal return to labor at time t of a plant created at time z should be equal to the wage.

Aggregation

Now we turn to the definition of the aggregate variables, the major strength of the Solow-like vintage capital based models being precisely its tractability in this respect.

Assuming that producing one unit of any capital good requires units of final good, gross investment at t is . Once defined an aggregate quality index

\[Q (t) = \int_ {0} ^ {M} q ^ {\kappa_ {j} (t) \tau \frac {1 - \alpha}{\alpha}} d i,\tag{5}\]

the production of the j-th capital good at t can be rewritten as follows using (3) and assuming that is constant for all j (which is true in equilibrium):

\[k _ {j} (t, t) = \frac {I (t)}{\eta Q (t)} q ^ {\kappa_ {j} (t) \tau \frac {1 - \alpha}{\alpha}}.\tag{6}\]

This equation simply establishes that the fraction of gross investment assigned to the j-th capital good depends mainly upon its relative eficiency. It is now possible to write aggregate counterparts for the production function and for the first order optimality conditions as well. Denote dz total employment. Using the first order condition (4), one can find an expression . If this expression is used in the definition of total employment given just above, one can find:

\[w (t) = \alpha B \left(\frac {K (t)}{L (t)}\right) ^ {1 - \alpha},\tag{7}\]

where is the eficient aggregate capital stock. Equation (7) restates the optimality condition (4) into the standard rule: The wage should be equal to the marginal productivity of labor. It can be rewritten in terms of the detrended capital stock per capita, as follows

\[\omega (t) = \alpha k (t) ^ {1 - \alpha}\]

where is the wage per eficiency unit, . Using (6), we can rewrite the eficient capital stock as the weighted sum of past and current investments, being the weights equal to the productivity of each investment:

\[K (t) = \int_ {- \infty} ^ {t} \frac {1}{\eta} Q (z) ^ {\frac {\alpha}{1 - \alpha}} I (z) e ^ {- \delta (t - z)} d z.\tag{8}\]

If we diferentiate (8), we obtain the accumulation law of eficient capital:

\[\stackrel {\bullet} {K} (t) = \frac {1}{\eta} Q (t) ^ {\frac {\alpha}{1 - \alpha}} I (t) - \delta K (t).\tag{9}\]

As in Solow (1960) and Greenwood, Hercowitz and Krusell (1997), the variation of the eficient capital stock is equal to eficient investment minus physical depreciation. There is, however, a crucial diference with respect to the previous contributions: The term , multiplying gross investment in the accumulation law, and as such interpretable as its marginal productivity or eficiency, is endogenous in our model.⁵ Using the accumulation law of eficient capital and the definition of detrended capital stock per capita, we can derive the dynamics of detrended eficient capital stock per capita:

\[\dot {k} (t) = \frac {1}{\eta} i (t) - \left(\delta + \frac {1}{1 - \alpha} g (t)\right) k (t),\tag{10}\]

where is detrended investment per capita, and (t) is the growth rate of the aggregate quality index.⁶

Before moving to the aggregation of the main equations of the model given our definition of aggregate capital, let us define an average age of capital variable, a variable which turns out to be crucial in our interpretation of the main mechanisms at work in this model as announced in the introduction. Within our set-up, the most natural way to define the average age of capital, m(t), is as follows:

\[m (t) = \frac {\int_ {- \infty} ^ {t} \frac {1}{\eta} (t - z) Q (z) ^ {\frac {\alpha}{1 - \alpha}} I (z) e ^ {- \delta (t - z)} d z}{K (t)}.\tag{11}\]

One can see that the integral appearing in the numerator is simply the “total of eficient capital given the vintage formulation of aggregate eficient capital given in equation (8). As it should be in this kind of settings, all vintages (ie. past and current investments) are weighted by their eficiency level in both the numerator and the denominator of the fraction giving .

We now turn to the aggregate relations implied by our model so as to show that it is possible to reinterpret the vintage model of the final goods sector in much more standard terms. This is pretty clear for equation (7) as mentioned above. Indeed, using (4), (7) and the definition of the aggregate production function, (2), one obtains after some easy algebra:

\[Y (t) = B L (t) ^ {\alpha} K (t) ^ {1 - \alpha},\tag{12}\]

L (t) = L

⁵This is a common characteristic with Krusell (1998).

⁶Equation (10) is obtained under the assumption for all t, which is true at equilibrium since labor supply is assumed constant. We keep on using this property hereafter.

while detrended output per capita is given by:

\[y (t) = B k (t) ^ {1 - \alpha}.\tag{13}\]

The first order condition (3) can be also rewritten in terms of the aggregate variables as:

\[(1 - \alpha) B \left(\frac {K (t)}{L (t)}\right) ^ {- \alpha} = P _ {K} (t) (r (t) + \delta + \xi (t) - \beta_ {K}),\tag{14}\]

where is the quality-adjusted price of capital goods in terms of the final good, and is the obsolescence rate of capital. This relationship is crucial in assessing the behaviour of the relative price of capital during the growth process, as it will be clear in the next section. As one can see, equation (14) simply states that the marginal productivity of capital should be equal to its user cost.⁷ Using the exact value of the monopoly price , we can rewrite the equation above in terms of detrended eficient capital stock per capita according to the following equation:

\[(1 - \alpha) B k (t) ^ {- \alpha} = \frac {\eta}{1 - \alpha} (r (t) + \delta + \xi (t) - \beta_ {K}).\tag{15}\]

The Schumpeterian research sector

How does the creative destruction process take place in the research sector? The story adopted here builds on the stochastic model of Aghion and Howitt (1992). A successful innovator increases the quality of the j-th capital good at t from grade to . The innovation is assumed to come out according to a Poisson probability scheme. Denote by , the Poisson arrival rate. We set: where is the flow of total resources (in terms of the final good) devoted to research in quality improvements of the j-th capital good at date t when the leading quality is We also assume that the Poisson arrival rate is a decreasing function of the research task, here captured by . Hence, . More concretely, we set , where λ is a positive parameter. The remaining term represents the negative efect of the complexity of the research task on the Poisson arrival rate.⁹ This choice is consistent, as we will see later, with an equilibrium Poisson probability ultimately independent of the complexity of the research task.

l (z, t)

⁷To obtain (14), an expression for is first derived thanks to (4), and we use it in (3). Then (7) is used to eliminate the wage variable from the modified (3). After some further simple algebra on the latter involving (6), we get our fundamental relation by diferentiation with respect to time.

⁸Since the final good is the unique input of the R&D activites, we are implicitly assuming that these activities use capital and labor according to the same technology as in the production of the final good. This hypothesis is quite standard in the literature, see Romer (1987), Rivera-Batiz and Romer (1991), Barro and Sala-i-Martin (1995, Chapters Howitt and Aghion (1998) and Evans 6, 7), et al. (1998).

⁹The same specification is considered in Barro and Sala-i-Martin (1995).

Let us assume that the marginal cost of producing any capital good of any quality ladder is the same, here equal to η units of the final good. The researcher responsible for each quality improvement retains a monopoly right to produce the good at the obtained quality. This is the usual story in research-based growth models where innovations are typically stimulated through some form of market power accruing to the innovators. Here, we add a markedly Schumpeterian flavor to the model. We assume that prior innovators, those responsible for the previous quality advances for the same capital good, lose automatically their monopoly rentals when a further quality improvement occurs.¹⁰ This feature generates a process of creative destruction, completely in line with the essence of Schumpeterian thought as previously figured out by Aghion and Howitt (1992).

Consistently with the properties above, we assume that for each capital good, all quality ladders are perfect substitutes as inputs in the production of the final good, and that . The latter condition is particularly crucial in generating mark-up pricing by the innovator and the elimination of the position of previous innovators. It basically states that the quality diferential between two successive innovations should be suficiently large. In this case, the above mentioned creative destruction process does take place.¹¹ In particular, the researcher responsible for the latest quality improvement retains a monopoly position and fixes a mark-up over its marginal cost given its demand. Rewriting equation (3), its demand is given by

\[(1 - \alpha) B q ^ {(1 - \alpha) (\kappa_ {j} (t) + 1) \tau} k _ {j} (t, t) ^ {- \alpha} \int_ {t} ^ {\infty} e ^ {- \int_ {t} ^ {z} (r (s) + \delta (1 - \alpha) - \beta_ {k}) d s} l (t, z) ^ {\alpha} d z = p _ {\kappa_ {j}} (t),\]

where the only diference with equation (3) is that the quality grade of the j-th capital good is now . The monopoly profit maximization implies that the mark-up should be constant, then the price of the j-th capital good is

\[p _ {\kappa_ {j}} (t) = p _ {\kappa} = \frac {\eta}{1 - \alpha}.\]

which is the same for all capital goods and all quality grades. From (6) follows that production of the -th capital good with quality grade , and the resulting flow of profits of the latest innovator is

\[\pi_ {\kappa_ {j}} (t) = \frac {\alpha}{1 - \alpha} \frac {I (t)}{Q (t)} q ^ {(\kappa_ {j} (t) + 1) \tau \frac {1 - \alpha}{\alpha}}\tag{16}\]

Indeed, a successful innovator is allowed to set a mark-up pricing when selling the demanded amount of the j-th capital good. Given the flow of profits’ expression (16), the value of an innovation improving the quality ladder of the j-th capital good at t is:

¹⁰We assume implicitly here that the researchers who occupy an incumbent position do no research. ¹¹More details on this standard condition and on the alternative cases can be found in Barro and Sala-i-Martin (1995), chapter 7. In the alternative case, the same results can be obtained ⁽¹ − ^{α) q} ≤ ^1, if Bertrand-like competition is assumed instead.

\[V _ {\kappa_ {j}} (t) = \frac {\alpha}{(1 - \alpha)} \int_ {t} ^ {\infty} \frac {I (t)}{Q (t)} q ^ {(\kappa_ {j} (t) + 1) \tau \frac {1 - \alpha}{\alpha}} e ^ {- \int_ {t} ^ {z} r (s) d s} e ^ {- \int_ {t} ^ {z} \gamma_ {\kappa_ {j}} (s) d s} d z,\tag{17}\]

where the two exponential terms in the integrand represent respectively the discount factor and the probability of the quality grade still leading at time We assume that the research sector is subsidized at a rate The arbitrage condition for the equilibrium level of resources spent in R&D activities should stipulate that the marginal cost of research is equal to or greater than the expected present value of profit with equality if the amount of resources devoted to R&D is strictly positive, that is:

\[1 - \beta_ {R} \geq \lambda q ^ {- (\kappa_ {j} (t) + 1) \tau \frac {1 - \alpha}{\alpha}} V _ {\kappa_ {j}} (t) \equiv \widetilde {V} _ {\kappa_ {j}} (t),\tag{18}\]

with equality if

Assuming that resources devoted to R&D are never nil, the arbitrage condition, (18), implies , which yields by diferentiation of (17)

\[V _ {\kappa_ {j}} (t) = \frac {\alpha}{1 - \alpha} \frac {I (t)}{Q (t)} \frac {q ^ {(\kappa_ {j} (t) + 1) \tau \frac {1 - \alpha}{\alpha}}}{r (t) + \gamma_ {\kappa_ {j}} (t)}.\tag{19}\]

From (18)-(19), it turns out that the Poisson arrival rate , , does not depend on the complexity of the research task, that is . This means that quality improvements can occur for all types of capital goods with the same probability and whatever is the reached quality grade. This property of the model is entirely due to the specification of function φ as outlined in Barro and Sala-i-Martin (1995), Chapter 7. In principle, the Poisson arrival rate is afected by in two opposite ways. First, the monopoly profits accruing to an innovator increase with since the amount of demanded capital good rises with the quality of the capital good as it transpires (6). Secondly, by assumption, the probability of innovating decreases with the dificulty of the task, measured by . When the specification is adopted, the two efects exactly ofset.

Assuming that resources devoted to R&D are never nil, and using (19), the arbitrage condition can be rewritten as

\[1 - \beta_ {R} = \frac {\lambda L \alpha}{(1 - \alpha)} \frac {s (t) y (t)}{r (t) + \gamma (t)},\tag{20}\]

where is the investment-output ratio:

\[s (t) \equiv \frac {I (t)}{Y (t)} = \frac {i (t)}{y (t)}.\]

It is worthwhile pointing out that the marginal return to innovation does depend on the investment to output ratio in our model. This is probably its most salient property. Such a property does not arise in the basic research-based growth models, both deterministic (Romer, 1990) or stochastic (Aghion and Howitt, 1992). In the latter setting, capital goods are not durable and hence there is no capital accumulation. However, incorporating capital in the creative destruction model as in Aghion and Howitt (1998), chapter 3, and Howitt and Aghion (1998) is not suficient to get the investment rate as a determinant of the profitability of R&D, as one can check. Indeed, this property is due to our vintage structure and to the underlying embodiment hypothesis. If technological progress were disembodied, the demand for the new designs of the capital goods would depend on total capital stock. In contrast, when technological progress is embodied, or in other words when technological progress is investment-specific, investment becomes the key variable of the profitability of the innovators’ activities.

The impact of the successive innovations on the aggregate quality index is derived in the usual way by applying the law of large numbers (see Aghion and Howitt, 1992, and Barro and Sala-i-Martin, 1995, chapter 7). In case of a quality improvement of the j-th capital good, the proportionate change in the quality grade is eAs established just above, the equilibrium Poisson probability of innovating is equal for all capital goods, which implies that the expected proportionate change in aggregate quality is . By the Law of Large Numbers, one can set the average growth rate of e(measured over any finite time interval) equal to the latter discrete proportionate change:

\[g (t) \equiv \frac {\stackrel {\bullet} {Q} (t)}{Q (t)} = \gamma (t) \widetilde {q}.\tag{21}\]

Closing up the model

The specification of the economy is completed by a standard modeling of consumers’ behaviour. We consider an infinitely lived representative household endowed with L units of labor. The household maximizes its intertemporal utility where is the time preference rate and is the inverse of the elasticity of intertemporal substitution. As usual, the optimality condition is

\[\frac {\stackrel {\bullet} {C} (t)}{C (t)} = \frac {1}{\sigma} (r (t) - \rho).\tag{22}\]

From the other hand, we assume that labor supply is inelastic and constant overtime, so that the clearing condition in the labor market writes simply

\[L (t) = L\tag{23}\]

Using (21) and (23), the Euler equation (22) can be rewritten as

\[\frac {\dot {c} (t)}{c (t)} = \frac {1}{\sigma} (r (t) - \rho) - g (t),\tag{24}\]

where is detrended consumption per capita.

Finally, the clearing condition in the market of the final good is

\[C (t) + I (t) + R (t) = Y (t),\tag{25}\]

where is the total amount of the final good devoted to research, Since the equilibrium Poisson probability is the same for all capital goods, , we get after some trivial algebra:

\[R (t) = \frac {\gamma (t)}{\lambda} (\widetilde {q} + 1) Q (t).\tag{26}\]

Using (13), (23) and (26), the clearing condition in the final good market can be rewritten, as:

\[c (t) + i (t) + \frac {\gamma (t)}{\lambda L} (\widetilde {q} + 1) = B k (t) ^ {1 - \alpha}.\tag{27}\]

We now turn to characterize the balanced growth equilibria of our model.

3 Characterizing balanced growth equilibria

We define a balanced growth path (BGP) as an equilibrium path along which the probability of successful innovation is constant, that is , while the growth rates of consumption, investment, resources devoted to R&D and output are kept also constant, equal to the rate of endogenous technological progress For their ecrucial economic relevance, we will focus more on the steady-states values of capital intensity (k), the Poisson arrival rate (γ), the rate of interest (r), the growth rate the obsolescence rate (ξ), the saving rate (s), and the average age of capital (m). These seven variables are simultaneously determined by the system:

\[s = \frac {(1 - \alpha) ^ {2}}{m (r + \xi + \delta - \beta_ {K})}\tag{S}\]

\[1 - \beta_ {R} = \frac {\lambda L B \alpha}{(1 - \alpha)} \frac {s k ^ {1 - \alpha}}{(r + \gamma)}\tag{G}\]

\[r = \sigma g + \rho\tag{R}\]

\[\gamma = \widetilde {q} ^ {- 1} g\tag{A}\]

\[\xi = \frac {\alpha}{1 - \alpha} g\tag{0}\]

\[m = \left(\frac {g}{1 - \alpha} + \delta\right) ^ {- 1}\]

\[k = \left(\frac {B s m}{\eta}\right) ^ {\frac {1}{\alpha}}\tag{M}\]

(K)

Equation (S) is obtained from (10), (13) and (15) together with (M) along the BGP, it states that the investment rate depends on the user cost of capital and the age structure of the capital stock as captured by the average age of capital. Equation (G) is the arbitrage condition in the R&D sector (20) along the BGP. Equation (R) is the familiar Fisher equation showing up how the interest rate depends on the growth rate of consumption. Equation (A) describes the relationship between the Poisson arival rate progress and the growth rate in the BGP. Equation (O) states that in the long run the obsolescence rate of capital is proportional to the growth rate. This relationship simply comes from the fact that the obsolescence rate is the rate of decline of the relative price of capital , which is equal by definition to Equation (M), corresponding to equation (11) in the BGP, implies that the average age of capital is a decreasing function of the growth rate in the long run. Finally, equation (K) expresses the steady-state value of capital intensity in terms of the long run investment rate and the long run average age of capital. It is obtained from (10) and (13) together with (M).

The most salient feature of our model is the crucial role of investment in the growth process. As mentioned before, our model predicts that the investment rate is a crucial determinant of the research efort as captured by the total amount of resources devoted to R&D (see equation (G)). This strategic complementarity may induce multiple equilibria, as it will be shown later. At the minute, observe that it implies that a rise in the investment rate stimulates innovation. More resources will be devoted to R&D, which increases the Poisson arrival probability together with the growth rate, and lowers the average age of capital by equation (M). A lower average age of capital induces a further rise in the investment rate via the equation (S), which in turn afects the growth rate and the average age of capital as before. And so forth. This is the the modernization mechanism of investment as outlined in the introduction. It derives fundamentally from our embodiment assumption.

It is worth pointing our here that our model presents some crucial diferences with respect to Howitt and Aghion (1998). As mentioned by these authors, their model combines the neoclassical growth model `a la Solow (1956) and their 1992 creative destruction model. Our framework is a combination of the latter and Solow’s vintage capital model (1960). There are essentially two diferences between the two approaches. At first, under disembodied technological progress and in contrast to the embodiment case, the obsolescence rate is zero and is not a determinant of the user cost of capital.

Second, and more importantly, the incentives to innovate depend on the total resources of the economy in Howitt and Aghion’s model, while they depend on the resources devoted to investment in ours. Indeed, one can check that the investment rate is not a determinant of the profitability of R&D in the former model, which instead depend on total capital stock. This diference crucially matters in the existence or not of the modernization efect as depicted above, and in the the way the economy respond to capital subsidies as we will show in sub-section 4.2.

As expected, our model is also able to reproduce the main empirical facts linked to embodiment, as pointed out previously by Greenwood, Hercowitz and Krusell (1997). The rate of decline of the relative price of capital and the growth rate of the ratio eficient capital to output are equal and more importantly, proportional to the rate of technological progress g. Indeed, by equation (10), the eficient capital per capita increases at a rate . As usual, due to embodiment, the growth rate of the capital stock, measured in eficiency units, is greater than the growth rate of output. The ratio eficient capital to output should consequently grow at a rate in the BGP, which is incidentally the rate of decline of the relative price of capital given by equation (O).

Last but not least, and as announced in the introduction, our vintage structure does also allow us to bring out several lessons as to optimal allocation of labor resources in the long run. Regarding the distribution of labor per plant, our model is very satisfactory. Indeed, along a BGP, the evolution of labor over time and its distribution across the plants is given by: . This equation is obtained by trivially combining equations (4), (5), (6), (7) and (M) in the BGP. Recall that is the amount of labor devoted to operate at t a plant constructed at In the BGP, as the plants get older, a lower amount of labour is assigned to them. Labor reallocation (namely creation plus destruction), being equal to is also found to increase when the growth rate g rises via the average age of capital variable. In this sense, our model does reproduce the most important stylized facts related to job creation and destruction as reported in Dunne et al. (1989), and Davis et al. (1996).

We now turn briefly to more technical considerations.

Existence and uniqueness of the balanced growth paths

Does the system (S)-(K) yield positive solutions for the targeted variables? Are the solutions unique? Do the positive solutions of this system imply positive values for the other variables of the system, namely consumption and investment? The following proposition gives simple suficient conditions under which unique positive solutions are ensured. To unburden the presentation and without any loss in the scope of our finding, we assume at the moment that the subsidy rates and are nil. The efects of these policy instruments on growth is reported in the next section.¹²

¹²Since the proof is simple and informative, we include in the main text.

Proposition 1 , where g solves the system , and where , there exists a unique solution to the system (S)- (K) with

Proof: The condition is needed to get utility bounded along the BGP for any . Moreover, condition is suficient to guarantee that consumption and investment are positive in the BGP. Let us prove it. From (S) and (K), it follows that

\[k = \left(\frac {(1 - \alpha) ^ {2} \frac {B}{\eta}}{(\sigma + \frac {\alpha}{1 - \alpha}) g + \delta + \rho}\right) ^ {\frac {1}{\alpha}},\tag{28}\]

k is strictly positive for all . From (27) and (A), we obtain

\[c = B k ^ {1 - \alpha} - i - \frac {\widetilde {q} + 1}{\widetilde {q} \lambda L} g.\tag{29}\]

From equation (K), using that , follows

\[k = \frac {B m}{\eta} i.\tag{30}\]

Using the equation just above, we can rewrite (29) as follows

\[c = \left(\frac {B m}{\eta} k ^ {- \alpha} - 1\right) i - \frac {\widetilde {q} + 1}{\widetilde {q} \lambda L} g.\tag{31}\]

On the other hand, the relations (G), (R) and (A), together with , imply

\[i = \frac {(1 - \alpha)}{\alpha \lambda L} \left(\left(\sigma + \frac {1}{\widetilde {q}}\right) g + \rho\right),\tag{32}\]

which implies that investment is positive in the BGP if the growth rate g is positive. Finally, substituting (28)and (32) into (31) yields the following fundamental relation

\[c = \frac {\widetilde {q} + 1}{\widetilde {q} \lambda L} g \left[ \left(m \frac {\left(\sigma + \frac {\alpha}{1 - \alpha}\right) g + \delta + \rho}{(1 - \alpha)} - (1 - \alpha)\right) \frac {(\widetilde {q} \sigma + 1) g + \widetilde {q} \rho}{\alpha (\widetilde {q} + 1) g} - 1 \right]\]

If then, using (M), and for all , we get

\[\begin{array}{l l} c & > \frac {(\widetilde {q} + 1) g}{\widetilde {q} \lambda L} \left[ \left(\frac {\frac {\alpha}{1 - \alpha} g}{g + \delta (1 - \alpha)} + \alpha\right) \frac {(\widetilde {q} \sigma + 1) g + \widetilde {q} \rho}{\alpha (\widetilde {q} + 1) g} - 1 \right] > \\ & > \frac {g}{\lambda L} \frac {\frac {1}{1 - \alpha} g}{g + \delta (1 - \alpha)} > 0. \end{array}\]

So under the conditions of Proposition 1, consumption is positive in the BGP as long as . To study the positivity of , some tedious algebra is needed. Indeed after successive substitutions involving equations (S), (G), (R), (A), (O), (M) and (K), we can write as an implicit function of the sole parameters of the problem:

\[1 = \frac {\Gamma \lambda L \left(\frac {1}{1 - \alpha} g + \delta\right)}{\left[ (\sigma + \widetilde {q} ^ {- 1}) g + \rho \right] \left[ \left(\sigma + \frac {\alpha}{(1 - \alpha)}\right) g + \delta + \rho \right] ^ {\frac {1}{\alpha}}} \equiv \widetilde {V} (g)\tag{33}\]

where . Equation (33) is obviously equation (G) expressed in terms of the unique endogenous variable In particular, the function represents the return to innovation as a function of e It is easy to check that function have the following properties: (i) (ii) the limit of is zero when tends to infinity, (iii) is continuous and strictly increasing in and (iv) there is at most one esuch that . From properties (i)-(iv) follows that for all there is only one strictly positive solution to (33). 2

Proposition 1 states, among other things, that the labor resources of the economy should be suficiently big for a BGP with positive growth rate to be sustainable. Though this kind of conditions is very often required in endogenous growth models (even in the simplest ones, see Romer, 1986), it is absolutely needed in our framework to additionally rule out multiplicity. Indeed, as explained in details above, our model presents a strategic complementarity between investment and R&D activities, and it is well known, since Cooper and John (1988) that multiple equilibria can be generated in such a case. In our model, a simple way to identify these features consists in studying the slope of . If this slope is positive (Resp. negative), an increase in the growth erate g rises (Resp. reduces) the return to innovation, which corresponds to strategic complementarities (Resp. substitutabilities).

In addition to the complementarity between investment and R&D activities, our model generates two strategic substitutabilities. The first one recovers an obsolescence efect. More resources devoted to R&D rises the discounting rate of the firms operating in the final goods sector, say via an increase in economic obsolescence. Indeed, where e e, by equation (O), is the obsolescence rate. Therefore, the marginal return to investment diminishes, and so will do successively investment and the marginal return to innovation, which will typically divert resources from the research sector. The second strategic substitutability comes from the well known business stealing efect. It is inherent to Schumpeterian models `a la Aghion and Howitt (1992), and as such it cannot be found in deterministic frameworks like Krusell’s model (1998) where the concept of planned obsolescence is more adapted. In our model, it works as follows: By construction of Poisson arrival rates, a rise in resources devoted to R&D reduces the expected lifetime of the monopoly power accruing to an innovator, and so it decreases the marginal return to innovation, which discourages in turn research efort.

So unsurprisingly, our model is able to generate multiple equilibria under certain conditions. However, we prefer to focus on the uniqueness case described by Proposition 1 for the following reason. Indeed, the BGP dominated by strategic complementarities turns out to arise only for suficiently low values of labor resources and more importantly, to be instable (in contrast to Young, 1993, for example).¹³ Moreover, the latter BGP arises for definitely unrealistic values of the parameters.¹⁴ Since the multiplicity of equilibria generated in our model is indeed fictitious, we abstract away from the related mathematical developments. Nonetheless, the idea that our model presents diferent sources of strategic complementarities and substitutabilities is appealing, and we still use it in the next comparative statics exercises.

Some comparative statics

Indeed, we perform some comparative statics exercises to complete the characterization of the BGP. Straightforwardly, we get the following results:

Proposition 2 Under the assumptions of Proposition 1, the steady state growth rate g depends positively on λ, q, L, and on B. On the contrary, g is negatively correlated with eand with η. The relationship between g and δ is ambiguous.

Proof: The proof is trivial. An increase in λ, L, B and rises for all Since the equilibrium growth rate is determined by ethanks to equation (33), ethe first part of the proposition follows immediately. The same argument applies for the second part since it can be trivially checked that an increase in and reduces for all . Things are more complicated with respect to the depreciation rate eδ. Indeed, note that the derivative of with respect to is:

\[\frac {\partial \widetilde {V}}{\partial \delta} = \frac {\Gamma \lambda L \left(1 - \frac {1}{\alpha} \frac {1}{\widetilde {r} m}\right)}{(r + \gamma) \widetilde {r} ^ {\frac {1}{\alpha}}}.\]

Hence the efect on a rise in is negative if

\[\frac {1}{\alpha} > \widetilde {r} m,\]

V0 (0) > 0,

L⁰ > 0

¹³One can check that if then there exists a value such that if ethree BGP arise, two of them with strictly positive growth rates g. Only one of the two relevant g. equilibria, namely the one dominated by strategic substitutabilities, turns out to be saddlepoint stable. All the details of the characterization of BGP’s existence and stability for any value of L can be found in del Rio (1999).

V0 (0) > 0,

¹−^αρ δ ³σ + q−1 + ^{σ+ α}1−αα(ρ+δ)

Λ < 1,

Λ =

Λ < 1

ρ = 0.02, δ = 0.05, α = 0.6

¹⁴For multiplicity to hold, we need which implies Λ < 1, where Λ = e´. Set ρ = 0.02, δ = 0.05, α = 0.6 and σ = 2, then Λ ≥ 85 ∀q ≥ 0! σ = 2 Λ ≥ 85 ∀q ≥ 0! eThe condition is impossible to obtain with acceptable parameters’ values.

and it is positive if

2

\[\frac {1}{\alpha} < \widetilde {r} m.\]

The comparative statics yield some interesting results. Note that an increase in λ rises the return to innovation by equation (18), and as such it should stimulates both innovation and growth. The same argument applies to the size of innovations parameter and to the marginal cost of capital goods (in the reverse direction of course). An eincrease in lowers the magnitude of the business stealing efect, which is good for egrowth. In contrast, an increase in is bad for the return to investment, which tends to discourage investment, thus afecting negatively the growth rate.

As usual in endogenous growth models, there is a scale efect: The greater is the size of the labor force, the larger will be output, and the more resources will be devoted to research. The same mechanism is present in the lab-equipment model of Rivera-Batiz and Romer (1991) for example. A similar scale efect is generated by the production function scale parameter The negative efect on growth of an increase in the impatience rate or in the preference parameter is abundantly documented in the literature. There are however two main departures with respect to Krusell (1998). He finds that the efect on growth of a rise in the physical depreciation rate, is ambiguous. Indeed, in his deterministic framework, an increase in also “rises the extent to which planned obsolescence hampers technological development” (Krusell, 1998, page 138), thus increasing the growth rate. In our stochastic Schumpeterian model this efect disappears. An increase in the depreciation rate has two opposite efects in our model. On one hand, it rises the user cost of capital, which discourages investment and growth; on the other hand, it lowers the average age of capital, which is positive for growth. Depending on the magnitude of each efect, the correlation between and can be positive or negative.

Another diference with respect to Krusell resides in the role of the number of capital goods M. Krusell finds that an increase in M should lower the growth rate through the following channel: Since his research sector uses labor, more varieties of capital goods means less labor available for each capital type, which implies in a symmetric equilibrium a lower growth rate. The same result is obtained by Young (1998). In our lab-equipment type modeling of the research sector, this property does not hold. Indeed, an increase in M requires more resources in the research sector exactly as in Krusell and Young. But the required increment is sustainable in our lab-equipment type model. And the rise in M will increase technological progress . In the BGP of our model, the cost involved by the presence of more varieties (more resources to operate the research sector) is sustainable and it is exactly ofset by the subsequent increment in technological progress. This explains why the long run growth rate does not depend on M in our model, in contrast to Krusell and Young.

We now turn to study the policy implications of our model in the BGP.

4 On the impact of subsidies and embodiment

In this section, we relax the assumption . We first present some insight into how the long run analysis is afected by this change. Then, we study more closely the efect of technological progress embodiment on the eficiency of capital subsidy, in connection with the discussion of Howitt and Aghion (1998).

4.1 The long run efects of capital and research subsidies

The conditions on the parameters for a BGP to exist are trivial extensions of the ones established in the benchmark case . A suficient condition for a unique BGP to exist is , which implies as before a lower bound for labor supply eL. Again, for suficiently high labor endowment L, only one BGP arises, and this BGP is dominated by strategic substituabilities. The efects of capital and research subsidy on growth turn out to be clear as reflects the following proposition.

Proposition 3 If the parameters of the model check the condition , the unique positive steady state growth rate g increases when either eor increase.

The proof is trivial and follows exactly the same arguments as in the proof of Proposition 2. Both subsidies stimulate growth in the long run. A rise in capital subsidy increases the marginal return to investment, and the main mechanism behind this is straightforward. An increase in reduces the discount rate of the firms in the final goods sector since in this case . This stimulates einvestment and research following the indirect modernization efect of investment. Indeed, an increase in tends to shift upward the marginal return to innovation, , ewhile the marginal cost of innovation is kept constant. On the other hand, an increase in research subsidy has the same expansive efects through a direct reduction in the marginal cost of research.

Therefore, subsidizing both capital or research has the same qualitative efects in the long run. As pointed out by Howitt and Aghion (1998), the traditional dichotomy between the accumulation of capital as a short run determinant of the growth process and R&D as the exclusive determinant of long run growth via technological advances does not make sense neither theoretically nor empirically. It is not relevant on the theoretical ground because extremely stringent conditions are required to generate such a property. Even if we abstract away from the embodiment hypothesis and the Arrowian learning by doing, capital accumulation is important for long run growth because capital is required for producing ideas on one hand, and for implementing and exploiting them on the other. Unless we hypothesize some extreme production functions in the research sector (as in Romer, 1990), the previous dichotomy cannot be generated.

Finally, note that our results are obtained on the BGP dominated by strategic substituabilities. The results would be inverted if a BGP dominated by strategic complementarities were to be considered. The reason for that has been already clearly identified by Young (1993): “...If one raises the return to an endogenous activity in a situation in which the return to economic actors is locally decreasing in their efort level, then a return to equilibrium will require an increase in their level of activity. However, if the return to the economic actors is locally increasing in their efort level, as is the case near the complementarity steady state, then a return to equilibrium requires a paradoxical reduction in their level of activity”. In our model the BGP dominated by strategic complementarities does arise for unrealistic parameters’ values and turns out to be instable. Therefore, the unique sustainable policy assessment lesson of our model is that both capital and research subsidy are growth improving.

4.2 Embodiment and the eficiency of capital subsidy

Aghion and Howitt (1998), chapter 3, argue that an increase in the capital subsidy has a lower (positive) efect on growth if obsolescence costs are to be considered. Precisely, the two authors compare a situation in which the obsolescence rate is zero due to a putty-putty specification (the non-embodiment case) with a situation in which a putty-clay technology yields a non-zero obsolescence rate equal to the innovations obsolescence rate, which is the Poisson probability in our model (say the emebodiment case). In both cases, a rise in capital subsidy stimulates growth. However, in the embodiment case, the subsequent drop in the growth rate induces an obsolescence efect, which afects negatively capital accumulation. This obsolescence efect specific to the embodiment case implies that capital subsidies are more eficient in terms of growth promotion in the non-embodiment configuration.

We argue here that Aghion and Howitt’s story is incomplete since it does not take into account the modernization efect of investment when technological progress is embodied. In our model, the embodiment assumption implies that the marginal return to innovation is positively correlated with the investment rate, and negatively correlated with the average age of capital. This is a characteristic of our truly vintage structure, which captures entirely the embodiment assumption in contrast to the putty-clay technology adopted by Aghion and Howitt. If the modernization efect is taken into account, then we have an additional ingredient with respect to the latter authors. The rise in investment due to the capital subsidy yields indeed a decline in the average age of capital, which again stimulates research and growth. This additional third effect may well rule out the main lesson from Aghion and Howitt’s model, namely the negative efect of embodiment on the eficiency of capital subsidy.

Let us give a very simple insight into this issue.¹⁵Let us consider the two following

¹⁵The treatment given below is just for an illustrative purpose. A much more rigorous appraisal of

equations characterizing the BGP of the model:

\[k ^ {- \alpha} = \frac {\eta}{(1 - \alpha) ^ {2} B} (\sigma g + \xi + \delta + \rho - \beta_ {K})\tag{S'}\]

\[1 - \beta_ {R} = \frac {\lambda L \alpha \eta}{1 - \alpha} \frac {k}{m (\sigma g + \rho + \widetilde {q} ^ {- 1} g)}\tag{\( (G') \}\]

eThe equation (S’) and (G’) are simply obtained by rewriting equations (S) and (G) using (K), once the assumption is relaxed. Recall that ξ and m represent the obsolescence cost and the average age of capital respectively. A higher growth rate g supposes an increase in ξ but a decline in m through the modernization efect of investment. How do these efects show up? In Figure 1, the system (S’)-(G’) is represented. The two curves intersect in a point O which coordinates are the long run values for the growth rate and capital intensity. For fixed ξ and if we rise the capital subsidy , the curve (S’) will shift upward, and we get a new equilibrium D with a higher growth rate. If technological progress were disembodied, the story would stop here. Under embodiment, ξ will rise and this obsolescence efect will imply a further downward shift in the curve (S’), with a new equilibrium point B. At B, the growth rate is lower than in D, which reflects exactly the argument of Aghion and Howitt. Now, if we account for the modernization efect, an increase in g will lower the average age of capital m, which shifts the curve (G’) upward, and we get a final equilibrium point I. At this point, the equilibrium point could be perfectly greater than at point O.

Our analysis traces back indeed to one of the most famous controversies in the sixties, the embodiment controversy (see Hercowitz, 1998, for a modern reincarnation of this debate). Those who supported the embodiment assumption argued that investment is the right channel through which innovations are implemented, and the capital stock is modernized, so that the investment rate should be a crucial determinant of long run growth. In a truly devastating paper, Phelps (1962) radically questioned this view. Within an exogenous growth framework with both embodied and neutral technological progress, he showed that while investment and its modernization efect are important for the pace of the growth process in the short run, they are not so in the long run: The distribution per vintage of the capital stock is ultimately independent of the total amount of capital and of the investment rate.¹⁶ Our model makes clear that this issue is far from simple in a research-based endogenous growth setting. In our vintage capital model, the growth rate is not independent of the average age of capital. An increase in investment stimulates research and growth, which lowers the average age of capital and induces a further expansion in investment and growth. In our context, the modernization efect is therefore a fundamental determinant of long run growth.

this question requires the accurate comparison of the exact efects of capital subsidy in our model and in a kind of “disembodied technological progress” counterpart. We think that our treatment is enough meaningful to make the point in economics terms without any need to any heavy complementary analytical developments.

¹⁶Denison (1964) denied any empirical relevance to the distinction between embodied and neutral technological progress. According to this contribution, embodiment is important if the modernization efect is. Through some simple (and questionable) accounting exercises, Denison found that this modernization efect is of a negligible magnitude. So he concluded for “the unimportance of the embodied question”.

5 Conclusion

In this paper, we have developed a general equilibrium model which incorporates both the creative destruction setup due to Aghion and Howitt (1992) and the vintage capital production function due to Solow (1960). In our view, this is the most natural way to introduce capital accumulation into the canonical creative destruction model. The most salient characteristic of our endogenous growth research-based model is that the profitability of R&D depends on the investment rate. This property is intimately related to the embodied nature of technological progress as we have shown. In such a case, the investment variable fully determines the demand for the new designs of capital goods, and the (expected) profits of the innovators. This generates a precise modernization mechanism via the investment rate and the average age of capital as depicted along the text. This modernization efect may well dominate the obsolescence costs inherent to embodiment, thus inducing a diferent assessment of the eficiency of capital subsidy in such a case. This aspect should be incorporated into the ongoing theoretical and empirical debate on the viability of the current IT boom as a long run growth regime.

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COLECCION RESUMENES

98-01: “Negociación colectiva, rentabilidad bursátil y estructura de capital en España”, Alejandro Inurrieta.

TEXTOS EXPRESS

2000-03: “Efectos sobre la inflación del redondeo en el paso a euros”, Mario Izquierdo y Simón Sosvilla-Rivero.

2000-02: “El tipo de cambio Euro/Dolar. Encuesta de FEDEA sobre la evolución del Euro”, Simón Sosvilla-Rivero y José A. Herce.

2000-01: “Recomendaciones para controlar el gasto sanitario. Otra perspectiva sobre los problemas de salud”, José A. Herce.

DOCUMENTOS DE TRABAJO

2000-27: “Obsolescence Vs modernization in a Schumpeterian vintage capital model”, Raouf Boucekkine, Fernando del Río y Omar Licandro.

2000-26: “Provisión de servicios públicos y localización industrial”, Luis Lanaspa, Fernando Pueyo y Fernando Sanz.

2000-25: “Labor Force Participation and Retirement of Spanish Older Men: Trends and Prospects”, Namkee Ahn y Pedro Mira.

2000-24: “Paridad del poder adquisitivo y provincias españolas, 1940-1992”, Irene Olloqui y Simón Sosvilla-Rivero.

2000-23: “Optimal Growth under Endogenous Depreciation, Capital Utilization and Maintenance Costs”, Omar Licandro, Luis A. Puch y J. Ramón Ruiz-Tamarit.

2000-22: “Expectativas, Aprendizaje y Credibilidad de la Política Monetaria en España”, Jorge V. Pérez-Rodríguez, Francisco J. Ledesma-Rodríguez, Manuel Navarro-Ibáñez y Simón Sosvilla-Rivero.

2000-21: “Población y salud en España. Patrones por género, edad y nivel de renta”, José Alberto Molina y José A. Herce.

2000-20: “Integration and Growth in the EU. The Role of Trade”, José A. Herce y Mª Luz García de la Vega.

2000-19: “Foreign Direct Investment and Productivity Spillovers”, Salvador Barrios.

2000-18: “Female Employment and Occupational Changes in the 1990s: How is the EU Performing Relative to the US?, Juan J. Dolado, Florentino Felgueroso y Juan F. Jimeno.

2000-17: “Do tobacco taxes reduce lung cancer mortality?, José Julián Escario y José Alberto Molina.

2000-16: “Solution to Non-Linear MHDS arising from Optimal Growth Problems”, J. R. Ruiz-Tamarit y M. Ventura-Marco.

2000-15: “El sistema de pensiones contributivas en España: Cuestiones básicas y perspectivas en el medio plazo”, Juan Francisco Jimeno.

2000-14: “Assessing the Credibility of the Irish Pound in the European Monetary System”, Francisco Ledesma-Rodríguez, Manuel Navarro-Ibáñez, Jorge Pérez-Rodríguez y Simón Sosvilla-Rivero

2000-13: “La utilidad de la econometría espacial en el ámbito de la ciencia regional”, Esther Vayá Valcarce y Rosina Moreno Serrano.

2000-12: “The role of the minimum wage in the welfare state: An appraisal”, Juan J. Dolado, Florentino Felgueroso y Juan F. Jimeno.

2000-11: “Modelling evolving long-run relationships: The linkages between stock markets in Asia”, José L. Fernández-Serrano y Simón Sosvilla-Rivero.