Optimal Growth under Endogeous Depreciation, Capital Utilization and Maintenance Costs by Omar Licandro* Luis A. Puch** J. Ramón Ruiz-Tamarit*** DOCUMENTO DE TRABAJO 2000-23

November, 2000

* FEDEA.

** Universidad Complutense.

*** Universidad de València.

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Optimal Growth under Endogenous Depreciation, Capital Utilization and Maintenance Costs∗

Omar Licandro FEDEA

Luis A. Puch U. Complutense

J. Ramón Ruiz-Tamarit U. València

November 2000

Abstract

This paper analyzes the equilibrium dynamics of an optimal growth model that incorporates endogenous depreciation, variable capital utilization, and expenditures on the maintenance of physical capital. Maintenance acts as a substitute for investment, since it reduces the depreciation of capital. Investment is subject to adjustment costs, and capital is not fully utilized, the degree of capital utilization afecting the activity of maintaining. We establish a set of suficient conditions for the existence and uniqueness of a steady state equilibrium. Also, we define a “delta golden rule” consistent with the proposed economic environment and we analyze the dynamic eficiency of this economy. Finally, the steady state is found locally saddle-path stable. These results provide a framework for the analysis of comparative dynamics in general equilibrium with these features.

Key words: Maintenance, Depreciation, Capital Utilization, Optimal growth.

JEL classification numbers: O40, E22, D90

∗Correspondence: J. Ramón Ruiz-Tamarit, Departament d’Anàlisi Econòmica, Universitat de València, E-46022 València, e-mail: ramon.ruiz@uv.es.

1 Introduction

Most analyses of aggregate economic activity take depreciation as exogenously given and ignore that equipment and structures have to be maintained and repaired. Moreover, an important margin along which a firm can adjust these activities has to do with the fraction of the installed capital stock being used. In this paper we develop a neoclassical growth framework that incorporates the endogenous determination of these variables. To this end, we augment the optimal growth model of saving and investment under adjustment costs introduced by Abel and Blanchard (1983), with a maintenance technology that acts as a substitute for investment and depends upon both the intensity with which physical capital is utilized and depreciates.

The exogenous nature of physical capital depreciation can be somewhat justified by considering a class of putty-putty models of production. However, it is well understood that this particular view of the capital accumulation process is quite restrictive. From the theoretical side, the assumption of exponential depreciation dramatically reduces the possible dynamics that an optimal growth model can describe. This is particularly relevant for growth theory as well as for investment theory. From the empirical side, the assumption of a constant depreciation rate turns out to be more in conformity with accounting principles than with those of economic theory. This is particularly important with respect to the measurement of physical capital.

There is evidence that the activity of maintaining and repairing equipment and structures [cf. McGrattan and Schmitz (1999)] is both large relative to investment and a substitute for investment to some extent. Furthermore, Licandro and Puch (2000) show that incorporating expenditures on the maintenance and repair of physical capital into models of aggregate economic activity will change the quantitative answers to some key questions that have been addressed with these models.¹ What it is missing is an analytical framework to characterize the equilibrium dynamics of the joint determination of investment rates, depreciation rates and utilization rates. Here, and this is the contribution of the paper, we give a first step in that direction by incorporating to the neoclassical growth framework a class of maintenance activities that are related with the capital ageing process and the decay that results from its use.

Our model specification builds upon previous results in Escribá-Pérez and Ruiz-Tamarit (1996) and Ruiz-Tamarit (1995). These authors explore the endogenous determination of depreciation under putty-putty technologies in partial equilibrium. In so doing, they introduce a maintenance activity that allows to reduce physical depreciation, which is positively related in turn with the intensity of use of capital under the depreciation-in use assumption.² We put at work these ideas into a general

¹See also Collard and Kollintzas (2000).

²Diferent specifications of this hypothesis have been discussed in Epstein and Denny (1980),

equilibrium framework.

The general equilibrium neoclassical growth model does not allow separation of the saving decisions of households from the investment decisions of firms. By introducing either a two sector technology [cf. Uzawa (1964)] or installation costs [cf. Abel and Blanchard (1983)] it is possible to overcome the essentially passive role of investment in the model. These analyses, however, rely on constant depreciation rates and full capacity utilization. Here we incorporate a maintenance technology into the standard growth model with adjustment costs. This technology allows us to augment the model to include endogenous depreciation and capital underutilization. Thus, under the necessary assumptions to derive well-defined investment, depreciation, and utilization functions, we characterize the steady state equilibrium, the short-run dynamics and the stability properties of our model.

As a result of our technological assumptions capital underutilization is optimal and the investment rate is determined simultaneously with the endogenous depreciation rate. Consequently, the equilibrium path is dynamically eficient. Furthermore, the long-run equilibrium of our economy is characterized by both optimal and golden rule capital accumulation that are below those of the standard neoclassical growth model with adjustment costs. Consequently, our technological assumptions suggest that a non-optimal depreciation policy, that ignores maintenance costs and variable utilization, might lead the economy to an excess of installed capacity.

In addition, we focus on the analysis of the dynamic properties of optimal trajectories. Once we prove local stability we present a set of numerical computations. We shall illustrate below that the presence of a simple maintenance costs technology reasonably can reduce the rate of convergence to the steady-state path. It turns out that these values of the speed of convergence are more in conformity with those supported by the empirical evidence. Also, along the optimal trajectories consumption, capital and output are positively related but the investment rate, defined as the ratio of investment over capital, is inversely related with them. Depreciation, utilization and maintenance rates are also inversely related with capital accumulation along the convergence path.

These findings provide a framework for the analysis of comparative dynamics in general equilibrium with these features. The rest of the paper is organized as follows. In Section 2 we introduce the model along with a discussion of the maintenance and adjustment costs technologies. In Section 3 we show existence and uniqueness of steady state equilibrium and we characterize optimal solutions. In Section 4 we present stability results. Section 5 concludes.

Bischof and Kokkelenberg (1987), Motahar (1992) and Burnside and Eichenbaum (1996). See also Rumbos and Auernheimer (1997) and the references therein.

2 The model and preliminary considerations

The goal here is to formalize the argument made above in the simplest optimal growth economy. In addition, we retain the assumption that investment is subject to an adjustment costs technology. The reason is twofold. On the one hand, this allows us to keep as a benchmark the specification in Abel and Blanchard (1983). On the other hand, the presence of a simple maintenance costs technology can be immediately justified in an economy where adjustment costs generate a well-defined investment demand function. We now introduce a general model economy of optimal growth with these features.

The economy is populated by a continuum of identical infinitely-lived households or dynasties that grow at an exogenously given rate . Each household discounts the future at a constant positive rate , and derives instantaneous utility from the consumption of an aggregate good, , according to , which is a mapping, increasing and strictly concave.

The technology is represented by a concave production function, increasing and linearly homogeneous in efective capital, , and labor, The fraction determines the intensity of use of the installed capital stock , thus . Indeed, N is population size. For simplicity of exposition we ignore the immediate extension to the case of exogenous technical progress of the labor augmenting type. Under the previous assumptions on ，

\[y _ {t} = f (k _ {t} u _ {t}),\tag{1}\]

where and are per capita output and per capita capital stock, respectively. Function is , increasing and strictly concave for all , lim and satisfies the Inada conditions.

Production may be allocated to consumption, the production of new capital goods, installation activities and maintenance services. Associated to these purchases are the two key ingredients of the present analysis, namely: investment is subject to an adjustment costs technology, and maintenance and repair are subject to a maintenance costs technology.

Let us assume that adjustment costs, which are internal to the firm, are represented by a linearly homogeneous function , increasing in gross investment, , and decreasing in the total capital stock, . Then where i is the rate of gross investment and is assumed , increasing and strictly convex for , with and . Consequently, per capita adjustment costs are then written as

By assumption, maintenance costs are internal to the firm and can be used to preserve capital goods from depreciation and use. These maintenance costs are represented by a linearly homogeneous function , decreasing in total depreciation and increasing in efective capital. Consequently, 2 where is the endogenous rate of depreciation. The function , the average maintenance costs, is assumed positive, , convex and linearly homogeneous on and . Furthermore, we assume and for [ and . The larger the utilization rate and the smaller the depreciation rate, the larger the maintenance costs of capital.³

The resource constraint is determined by the following equalities

\[c _ {t} + (i _ {t} + \phi (i _ {t}) + m (\delta_ {t}, u _ {t})) k _ {t} = f (k _ {t} u _ {t})\tag{2}\]

\[\dot {k} _ {t} = (i _ {t} - \delta_ {t} - n) k _ {t},\tag{3}\]

where denotes the time derivative of per capita physical capital with respect to time.

In the present setting, every optimal solution may be decentralized as a competitive equilibrium. Thus, without loss of generality we shall confine our analysis to the planner’s problem. The planner’s optimization problem is to choose at each moment in time the rates of investment, depreciation and capital utilization so as to maximize the infinite stream of discounted instantaneous utilities, given the resource constraints (2) and (3), and the initial capital stock,

Definition 1 An optimal solution for this economy is a set of paths 2 , for t positive, which solve

\[\max \int_ {0} ^ {\infty} U (c _ {t}) e ^ {- (\beta - n) t} d t,\tag{P}\]

subject to and given, and where all variables are assumed to be strictly positive and strictly smaller than one.

Definition 2 A steady-state equilibrium for this economy is an optimal solution to problem , such that and remain constant.

It is readily shown that not only rates but also equilibrium per capita variables remain constant at steady state. Consequently, at a steady state the equilibrium levels grow at the rate n.

³An equivalent representation of the problem can be achieved by assuming that the depreciation rate is a function of the utilization rate and the rate of maintenance costs to capital.

An interior optimal solution to problem (P) must satisfy, dropping time subscripts, the following first order equation system

\[\mu = U ^ {\prime} (c) \left(1 + \phi^ {\prime} (i)\right)\tag{4}\]

\[\mu = - U ^ {\prime} (c) m _ {\delta} (\delta , u)\tag{5}\]

\[f ^ {\prime} (k u) = m _ {u} (\delta , u)\tag{6}\]

\[\dot {\mu} = U ^ {\prime} (c) (i + \phi (i) + m (\delta , u) - f ^ {\prime} (k u) u) + \mu (\beta + \delta - i)\tag{7}\]

\[\dot {k} = (i - \delta - n) k\tag{8}\]

\[c + k (i + \phi (i) + m (\delta , u)) = f (k u)\tag{9}\]

given and the corresponding transversality condition

\[\lim _ {t \to \infty} \mu_ {t} k _ {t} \mathrm{e} ^ {- (\beta - n) t} = 0.\tag{10}\]

The multiplier represents the shadow price of an additional unit of installed capital. The term is the marginal opportunity cost of gross investment. Then, equation (4) states that this marginal cost must be equal to the shadow price of capital. On the other hand, is the marginal saving in maintenance costs associated to an increase in the depreciation rate. An increase in δ reduces the capital stock and, hence, maintenance expenditures. equation (5) states that this marginal saving must be equal to the shadow price of capital. The term is the marginal maintenance cost associated to an increase in the utilization rate. Equation (6) states that this marginal cost must be equal to the marginal productivity of such an increase in the utilization rate, measured by the term u).

From (4) to (6) and (9), we can write and . Additionally, combining (4) to (7) and the assumption that is linearly homogeneous, the dynamic system (7) and (8) can be written in the phase space as

\[\frac {\dot {\mu}}{\mu} = \beta - H (i (k, \mu)) \equiv \frac {\Gamma (k , \mu)}{\mu}\tag{11}\]

\[\frac {\dot {k}}{k} = i (k, \mu) - \delta (k, \mu) - n \equiv \frac {\Pi (k , \mu)}{k},\tag{12}\]

where

\[H (i) = \frac {i \phi^ {\prime} (i) - \phi (i)}{1 + \phi^ {\prime} (i)}.\]

Indeed, summarizes all the marginal efects on the return to capital. From the assumed properties of function , we can easily prove that and , for all . Moreover, li and from l’Hôpital rule and after some elementary calculations, lim . In order to analyze this dynamic system, we first show existence and uniqueness of a steady state and then local stability.

3 Characterization of steady state solutions

The above optimization problem difers from the standard optimal growth model with adjustment costs because of the presence of maintenance costs and the underutilization of capital. Therefore, before discussing the properties of the optimal steady state, we establish its existence and uniqueness in Proposition 1. We also define a golden rule for this economy, that we call delta golden rule, and we study the dynamic eficiency of the unique steady state solution and its relation with those corresponding to the benchmark neoclassical framework.

3.1 Existence and uniqueness of steady state solutions

The following proposition establish suficient conditions for the existence and uniqueness of a steady state equilibrium.

Proposition 1 Under the following limit conditions:

\[i) | \lim _ {u \to 0 ^ {+}} m _ {\delta} (\delta , u) | < 1 + \phi^ {\prime} (n + \delta), f o r a l l \delta > 0\]

\[i i) \left| \lim _ {u \rightarrow 1 ^ {-}} m _ {\delta} (\delta , u) \right| > 1 + \phi^ {\prime} (n + \delta), f o r a l l \delta > 0\]

An interior steady state exists and it is unique.

Proof. From equation (11) at steady state. Given that , and lim , we may easily conclude that for any there is only one positive value for the investment rate,

Then, from (12) the steady state depreciation rate . Given that and , which implies

We combine (4) and (5) to obtain . From the right hand side of this equation is increasing in u. Given i and δ, conditions i) and ii) are suficient for the existence of a unique solution for

From (6), . Given that δ and u are interior at steady state, is a strictly positive finite number. From the Inada conditions imposed on function , there exists one and only one interior steady state value for .

A steady state value for c can be obtained from (9), and the existence and uniqueness of a finite solution for it can be easily verified, given our assumptions on functions and . To prove positivity, let us combine (9) with the other optimal conditions and the assumption of linear homogeneity of and get

\[c = \underbrace {f (k u) - f ^ {\prime} (k u) k u} + k (1 + \phi^ {\prime} (i)) (H (i) - n) > 0\]

Given our assumptions on function f(.), the first term on the right hand side is positive. At steady state, , which implies that the last term is also positive.

From (4), an interior solution for exists and is unique.

3.2 The (modified) delta golden rule

In our framework, the feasibility constraint at steady state can be written as

\[c = f (k u) - k [ \delta + n + \phi (\delta + n) + m (\delta , u) ].\]

Of course, the degree of capital utilization and the depreciation rate are not exogenously given. In order to make intertemporal eficiency comparisons we define a delta golden rule.

Definition 3 The delta golden rule is the value of k consistent with the maximization of steady state consumption with respect to k, δ and u, i.e., the solution of the following system:

\[f ^ {\prime} (k u) u = \delta + n + \phi (\delta + n) + m (\delta , u)\tag{13}\]

\[f ^ {\prime} (k u) = m _ {u} (\delta , u)\]

\[1 + \phi^ {\prime} (\delta + n) = - m _ {\delta} (\delta , u).\]

Notice that the last two equations in Definition 3 are equal to equations (6) and , respectively. In the following proposition, we show that an interior delta golden rule exists and is unique.

Proposition 2 Under conditions i) and ii) of Proposition 1 together with 2 there is a unique delta golden rule.

Proof. Combining the three equations in Definition 3, we

\[H (\delta + n) = n.\]

Provided that , since , and , there exists one and only one satisfying Definition 3.

For a finite the left hand side of the last equation in Definition 3 is constant. The right hand side is increasing in since . Then, conditions i) and ii), are suficient to the existence of a unique solution for u

Given a finite and , the right hand side of the first equation in Definition 3 is finite. From and the Inada conditions, one and only one interior solution for k does exist.

For obvious reasons, any steady state value for the per capita capital stock that exceeds the delta golden rule value, denoted , is dynamically ineficient irrespective of the corresponding values for and u. Of course, the steady state of our model economy, denoted , is optimal and verifies that . For further comparative analysis it is convenient to express equation (11) at steady state equilibrium values

\[f ^ {\prime} \left(k ^ {*} u ^ {*}\right) u ^ {*} = \delta^ {*} + \beta + \phi \left(\delta^ {*} + n\right) + m \left(\delta^ {*}, u ^ {*}\right) + (\beta - n) \phi^ {\prime} \left(\delta^ {*} + n\right),\tag{14}\]

We call this equation the modified delta golden rule, to distinguish it from the modified golden rule of the Ramsey-Cass-Koopmans model. We should note from (14) that determination of the optimal capital stock requires the simultaneous determination of all control variables. Consequently, there are important sources of variation in steady-state solutions related to changes in the parameters of the maintenance and adjustment costs technologies. This is an important implication of our model specification that goes beyond the somewhat counterfactual diferences in preferences, population growth and technical progress the standard model needs to account for income diferences across countries.

The following proposition shows the relation between the modified delta golden rule and the delta golden rule for some key variables.

Proposition 3 The following relations between the modified delta golden rule and the delta golden rule must hold: and

Proof. The golden rule implies , while at steady state . Since and is monotonically increasing, Thus,

Moreover, given and , we also conclude that

For both the golden and the modified golden rule, Given that and , then , implying that . Now, given . The previous statements imply that , and the comparison between the two extreme terms, given , says that

By definition, the golden rule implies . Using the aggregate resource constraint, we get

\[f (k _ {g} u _ {g}) - (\delta_ {g} + n + \phi (\delta_ {g} + n) + m (\delta_ {g}, u _ {g})) k _ {g} \geq\]

\[\geq f \left(k ^ {*} u ^ {*}\right) - \left(\delta^ {*} + n + \phi \left(\delta^ {*} + n\right) + m \left(\delta^ {*}, u ^ {*}\right)\right) k ^ {*}.\]

Then, using equation (13) and (14), and rearranging terms we obtain the following inequality:

\[\begin{array}{r c l} f (k _ {g} u _ {g}) - f ^ {\prime} (k _ {g} u _ {g}) k _ {g} u _ {g} & \geq & f (k ^ {*} u ^ {*}) - f ^ {\prime} (k ^ {*} u ^ {*}) k ^ {*} u ^ {*} + (\beta - n) [ 1 + \phi^ {\prime} (\delta^ {*} + n) ] k ^ {*} > \\ & > & f (k ^ {*} u ^ {*}) - f ^ {\prime} (k ^ {*} u ^ {*}) k ^ {*} u ^ {*}. \end{array}\]

So, given the strict concavity of the production function we may easily conclude that . Then, given , it becomes obvious that

From the previous result,

Thus, a higher productivity of physical capital in the long-run is associated with long-run levels of the depreciation rate, the investment rate and the utilization rate that are above those of the golden-rule solution. This result is standard in the optimal growth literature. Less immediate results show up through comparison with the Ramsey-Cass-Koopmans and the Abel-Blanchard models.

Indeed, the equilibria of the Ramsey-Cass-Koopmans model and that of Abel and Blanchard are characterized by

\[f ^ {\prime} (k _ {R}) = \delta + \beta\tag{15}\]

and

\[f ^ {\prime} (k _ {A}) = \delta + \beta + \hat {\phi} (\delta + n) + (\beta - n) \hat {\phi} ^ {\prime} (\delta + n)\tag{16}\]

respectively, where the utilization rate is supposed to be equal to one and the depreciation rate is an exogenous parameter. Under standard assumptions on the adjustment cost function, should be positive, which implies that . In Proposition 4, we compare the Abel and Blanchard steady state equilibrium with our delta and modified delta golden rules. In order to do this comparison, we assume that , where , and . The rationale for these assumptions is the following. In our model, maintenance and repair are considered separate economic activities, but in Abel and Blanchard, investment includes them. Consequently, in our model depreciation is net of maintenance, but not in Abel and Blanchard. For the same reason, we must renormalize the investment function: it depends on gross investment in Abel and Blanchard and on investment net of maintenance in our model.

Proposition 4

Proof. Since and , from (14) and (16) we get . Consequently, and

\[\frac {k _ {A}}{k ^ {*}} = \frac {f ^ {\prime} (k _ {A}) k _ {A}}{f ^ {\prime} (k ^ {*} u ^ {*}) k ^ {*} u ^ {*}}.\]

Let us define . We can easily proof that, for . Then,

\[\frac {f ^ {\prime} (k _ {A}) k _ {A}}{f ^ {\prime} (k ^ {*} u ^ {*}) k ^ {*} u ^ {*}} > 1 \quad i f \quad \Theta (k) < 1,\]

which completes the proof.

Θ(k) represents the curvature of the production technology . For a CES production function with elasticity of substitution larger than one, this assumption holds for any . This result can also hold for a CES production function with gross complementarity, provided that is not too large, since independently of the elasticity of substitution.

The intuition behind the result in Proposition 4 is straightforward. A nonoptimal depreciation policy, that ignores variable maintenance costs and utilization, leads the economy to an excess of installed capacity. However, this excess of capacity is not necessarily dynamically ineficient.

4 Dynamic analysis of optimal trajectories

In Proposition 5 we prove local stability of the unique interior steady state equilibrium.

Proposition 5 The unique interior steady state equilibrium of the dynamic system and (12) is locally saddle-path stable.

Proof. A first order Taylor expansion of (11) and (12) may be written in matrix form as

\[\binom{\dot {k}}{\dot {\mu}} = \left( \begin{array}{c c} \Pi_ {k} (k ^ {*}, \mu^ {*}) & \Pi_ {\mu} (k ^ {*}, \mu^ {*}) \\ \Gamma_ {k} (k ^ {*}, \mu^ {*}) & \Gamma_ {\mu} (k ^ {*}, \mu^ {*}) \end{array} \right) \binom{k - k ^ {*}}{\mu - \mu^ {*}},\]

where denotes the steady state value of x. As it is shown in the Appendix, the coeficients of the Jacobian matrix are:

\[\Pi_ {k} (k ^ {*}, \mu^ {*}) = k ^ {*} (i _ {k} ^ {*} - \delta_ {k} ^ {*}) > 0\]

\[\Pi_ {\mu} (k ^ {*}, \mu^ {*}) = k ^ {*} \left(i _ {\mu} ^ {*} - \delta_ {\mu} ^ {*}\right) > 0\]

\[\Gamma_ {k} (k ^ {*}, \mu^ {*}) = - \mu^ {*} H ^ {\prime} (i ^ {*}) i _ {k} ^ {*} > 0\]

\[\Gamma_ {\mu} (k ^ {*}, \mu^ {*}) = - \mu^ {*} H ^ {\prime} (i ^ {*}) i _ {\mu} ^ {*} < 0\]

Following Kurz (1968), the trace of the Jacobian matrix must be:

\[\mathrm{trace} J ^ {*} = \gamma_ {1} + \gamma_ {2} = k ^ {*} [ i _ {k} ^ {*} - \delta_ {k} ^ {*} ] - \mu^ {*} H ^ {\prime} (i ^ {*}) i _ {\mu} ^ {*} = \beta - n > 0,\]

where and are the eigenvalues. On the other hand, the determinant of the Jacobian matrix is:

\[\det J ^ {*} = \gamma_ {1} \cdot \gamma_ {2} = \mu^ {*} H ^ {\prime} (i ^ {*}) k ^ {*} [ i _ {\mu} ^ {*} \delta_ {k} ^ {*} - i _ {k} ^ {*} \delta_ {\mu} ^ {*} ] < 0\tag{17}\]

Consequently, the two eigenvalues are, respectively, 0 and . These features of the Jacobian matrix mean that the system has a saddle point dynamical structure. , given the initial condition for the predetermined variable and the transversality condition (10), the system places on the stable arm and then converges to the steady state equilibrium. Otherwise the system explodes.

In particular, given convergence implies that increases but decreases because . The speed of convergence, measured as the absolute value of the negative eigenvalue, is given by , with

It seems dificult to state general conditions characterizing the speed of convergence in our model. To further examine the dynamic properties of the convergence path we resort to numerical computations. First, we investigate the impact of our technological assumptions in the speed of convergence of the neoclassical growth model. The following specific functional forms are used throughout: and

Table 1: Efect of parameters b and α on steady-state values and the rates of convergence

We consider our model, together with Abel and Blanchard model under the corresponding interpretations of the depreciation rate and the investment rate as discussed above. We fix parameters and , the other parameters varying as specified in Table It is worth pointing out that parameter d of the maintenance costs function afects the steady-state values , and but not the rate of convergence. In this table, is the speed of convergence of the Abel and Blanchard model.

The main conclusion that can be drawn from Table 1 is that the rate of convergence in our model is substantially reduced compared with the standard neoclassical growth model with adjustment costs. Indeed, the decrease in the speed of convergence we obtain assigns more importance to the transitional dynamics of our model. This result is in line with related literature that shows the importance of variable utilization rates of capital in shaping the saddle path and the convergence rate. For instance, Rumbos and Auernheimer (1997) quantitatively compare the rate of convergence of small open model economies with fix and variable utilization rates. They find slower convergence under variable utilization, the absolute diferences going up to 18 per cent. Here we have a lot more of intratemporal substitution between investment and maintenance through a variable utilization rate and an endogenous depreciation rate so much so that convergence is at least twice as fast in the standard neoclassical model with adjustment costs. Furthermore, for our baseline economy with and we obtain a rate of convergence which is more in conformity with those reported in some empirical studies (Barro and Sala-i-Martin (1992) report annual rates of convergence of the order of 2 per cent).

Finally, Table 2 summarizes the dynamic behavior of the variables in our model along the capital accumulation convergence path for alternative values of b and α. In all our numerical experiments consumption and investment reacts as in the standard

b = 10,

q (1.5).

u = 1, ν_A = 2%

⁴The benchmark value of the adjustment cost coeficient, is chosen to get a plausible value for Tobin’s For the benchmark value of the capital elasticity, α = 0.40, we consider α = 0.40 variations in parameter b such that and respectively. ν = 2%,

Table 2: Efect of parameters b and α in the dynamic behavior of variables along the convergence path

neoclassical model.

As the initial stock of capital is below its steady state value, the optimal initial reaction is to accumulate, to maintain and to utilize capital at higher rates than in steady state. For this reason, on the adjustment process all these rates are decreasing. However, the depreciation rate does not initially decrease, but increases. This depends crucially on the negative sign of the cross derivative in the maintenance function: from (5), utilization and depreciation move in the same direction.

5 Conclusions

In this paper we introduce maintenance costs, endogenous depreciation and capital utilization rates in an otherwise optimal growth model with investment adjustment costs. Our model specification generalizes that of Abel and Blanchard (1983) and provides a theoretical framework for the analyses of comparative dynamics in a class of general equilibrium models with capital underutilization and maintenance activities.

The social optimum is characterized by an endogenous simultaneous determination of the investment rate, the depreciation rate and the utilization rate. This circumstance has some relevant implications. First, in steady state capital is optimally underutilized and maintenance activities are optimally undertaken. Second, we define a delta golden rule and we show that the steady state equilibrium of our economy verifies a modified delta golden rule, which is dynamically eficient. Finally, we show that the unique steady state equilibrium is locally stable.

But we have taken only one necessary step in characterizing the equilibrium dynamics of a growth model with endogenous depreciation, capital underutilization and spending on maintenance and repair. Further work, in line with Licandro and Puch (2000), is needed to definitively establish the extent to which including these features in aggregate models will change the answers to quantitative questions.

References

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Appendix: Control functions and its partial derivatives at steady state

Taking equations , and the production function in intensive form , we can implicitly define the following optimal control functions: , and . Via the implicit function theorem we can identify the corresponding partial efects, evaluated at the steady state where

\[\begin{array}{r l r} {u _ {k}} & {=} & \frac {f ^ {\prime \prime} u \{[ U ^ {\prime} ] ^ {2} \phi^ {\prime \prime} m _ {\delta \delta} - U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2} U ^ {\prime} m _ {\delta \delta} - U ^ {\prime} \phi^ {\prime \prime} U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2} \}}{\Delta} - \\ & & {- \frac {[ \beta - n ] U ^ {\prime} \phi^ {\prime \prime} m _ {u \delta} [ m _ {\delta} ] ^ {2} U ^ {\prime \prime}}{\Delta} < 0} \\ {u _ {\mu}} & {=} & {\frac {U ^ {\prime} \phi^ {\prime \prime} m _ {u \delta}}{\Delta} < 0} \\ {i _ {k}} & {=} & {\frac {U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2} U ^ {\prime} f ^ {\prime \prime} m _ {\delta \delta} [ \beta - \delta - n ]}{\Delta} < 0} \\ {i _ {\mu}} & {=} & {- \frac {U ^ {\prime} f ^ {\prime \prime} k m _ {\delta \delta}}{\Delta} > 0} \\ {\delta_ {k}} & {=} & {- \frac {U ^ {\prime} m _ {u \delta} f ^ {\prime \prime} u \{U ^ {\prime} \phi^ {\prime \prime} - U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2} \} - [ m _ {\delta} ] ^ {2} U ^ {\prime \prime} [ \beta - n ] U ^ {\prime} \phi^ {\prime \prime} [ m _ {u u} - f ^ {\prime \prime} k ]}{\Delta} < 0} \\ {\delta_ {\mu}} & {=} & {- \frac {U ^ {\prime} \phi^ {\prime \prime} [ m _ {u u} - f ^ {\prime \prime} k ]}{\Delta} < 0} \\ {c _ {k}} & {=} & {\frac {m _ {\delta} [ \beta - \delta - n ] f ^ {\prime \prime} k [ U ^ {\prime} ] ^ {2} m _ {\delta \delta} \phi^ {\prime \prime}}{\Delta} < 0} \\ {c _ {\mu}} & {=} & {- \frac {k m _ {\delta} \{U ^ {\prime} f ^ {\prime \prime} k m _ {\delta \delta} - U ^ {\prime} \phi^ {\prime \prime} [ m _ {u u} - f ^ {\prime \prime} k ] \}}{\Delta} < 0} \\ {y _ {k}} & {=} & \frac {f ^ {\prime} u \Delta + f ^ {\prime} k f ^ {\prime \prime} u \{[ U ^ {\prime} ] ^ {2} \phi^ {\prime \prime} m _ {\delta \delta} - U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2} U ^ {\prime} m _ {\delta \delta} - U ^ {\prime} \phi^ {\prime \prime} U ^ {\prime \prime} k [ m _ {\delta} ] ^ {2}}{\Delta} - \\ & & {- \frac {f ^ {\prime} k [ \beta - n ] U ^ {\prime} \phi^ {\prime \prime} m _ {u \delta} [ m _ {\delta} ] ^ {2} U ^ {\prime \prime}}{\Delta} > 0} \\ {y _ {\mu}} & {=} & {\frac {f ^ {\prime} k U ^ {\prime} \phi^ {\prime \prime} m _ {u \delta}}{\Delta} < 0} \end{array}\]

where . Finally, after some elementary manipulations, we obtain that

COLECCION RESUMENES

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

TEXTOS EXPRESS

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.

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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.

2000-10: “Integration and Inequality: Lesson from the Accessions of Portugal and Spain to the EU”, Juan F. Jimeno, Olga Cantó, Ana Rute Cardoso, Mario Izquierdo y Carlos Farinha Rodrigues.

2000-09: “Explaining Youth Labor Market Problems in Spain: Crowding-Out, Institutions, or Technology Shifts”, Juan J. Dolado, Florentino Felgueroso y Juan F. Jimeno.

2000-08: “Distributional aspects of the quality change bias in the CPI: Evidence from Spain”, Javier Ruiz-Castillo, Eduardo Ley y Mario Izquierdo.

2000-07: “Testing chaotic dynamics via Lyapunov exponents”, Fernando Fernández-Rodríguez, Simón Sosvilla-Rivero y Julián Andrada-Félix.