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Endogenous Growth, Capital Utilization and Depreciation by J. Aznar-Márquez* J. R. Ruiz-Tamarit** DOCUMENTO DE TRABAJO 2004-21

October 2004

* Universitat Miguel Hernández d’Elx (Spain). E-mail: juana.aznar@umh.es ** Corresponding author. Universitat de València (Spain) and IRES, Université Catholique de Louvain (Belgium). Departament d’Anàlisi Econòmica, Avda. dels Tarongers s/n, E-46022 València (Spain). Phone: +34 963828250; Fax: +34 963828249; E-mail: ramon.ruiz@uv.es

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Endogenous Growth, Capital Utilization and Depreciation∗

J. Aznar-Márquez† J. R. Ruiz-Tamarit‡

Revised, september 2004.

Abstract

We study an extended version of the one-sector AK growth model introducing adjustment and maintenance costs. Agents are allowed to under-use the installed capital and to vary the depreciation rate. The model is analyzed using particular functional forms and is solved in closed-form. We find that adjustment and maintenance costs (eficiency) reduce (increases) investment, depreciation, capital utilization and the rate of growth; impatience reduces the rate of growth but increases depreciation and utilization, which are also negatively related to the rate of population growth; the rate of growth appears positively correlated with the depreciation rate and the rate of capital utilization.

Keywords: Maintenance, Depreciation, Capital Utilization, Endogenous Growth.

JEL classification: O40, E22, D90.

Running title: Maintenance and Growth.

∗We thank R. Boucekkine and O. Licandro for their helpful comments. Authors acknowledge the support of the Belgian research programme ARC 03/08-302 and the financial support from the Spanish CICYT, Projects SEC99-0820 and SEC2000-0260. Ruiz-Tamarit also acknowledges the grant PR2003-0107 from the Secretaría de Estado de Educación y Universidades, Spanish MECD.
†Universitat Miguel Hernández d’Elx (Spain). Avda. del ferrocarril s/n, Edifici Galia, E-03202 Elx (Spain). E-mail: juana.aznar@umh.es
‡Corresponding author. Universitat de València (Spain) and IRES, Université Catholique de Louvain (Belgium). Departament d’Anàlisi Econòmica, Avda. dels Tarongers s/n, E-46022 València (Spain). Phone: +34 963828250; Fax: +34 963828249; E-mail: ramon.ruiz@uv.es

1 Introduction

In the standard neoclassic growth model, the decision about how much to save is based on the comparison, in welfare terms, between the costs and the benefits of a higher consumption today rather than tomorrow. Once the amount of savings has been decided, they are automatically channelled into investment. Therefore, investment has an entirely passive role. In such a model it is assumed full utilization of the installed capital and that depreciation experienced by capital equipment is an exogenously determined constant fraction of capital stock. These assumptions, however, do not conform to observed facts because available data show quite a diferent reality. Firms do not always decide to use all the installed capital and they are able to influence on the depreciation rate of capital stock. The way the firms may act on the depreciation rate is by devoting resources to the preservation, that is repair and maintenance, of capital stock which has deteriorated either through the use in the production process or simply through the natural process of ageing.

Recently, McGrattan and Schmitz (1999) highlighted the quantitative relevance of repair and maintenance activities. These authors obtain in Canada for the period 1961-1993 that up to 6% of gross national product was devoted to capital repair and maintenance, which is approximately half the expenditure made on the acquisition of new capital goods. In addition, Gylfason and Zoega (2001), using data currently published by the World Bank, studies the relationship between depreciation and growth. Among the main results, we would like to emphasize the following ones: i) increased population growth accelerates depreciation, ii) increased eficiency increases depreciation, and iii) increased long-run growth also accelerates depreciation. Furthermore, they document an important positive correlation between the rate of growth of per capita income and the depreciation rate over the period 1965-1998 for a sample of 85 countries.

Despite the above-mentioned empirical evidence, the depreciation rate has been regarded as an exogenous parameter in growth theory which, in the case of neoclassical models, negatively afects long-run variable levels and short-run growth rates. Moreover, in the case of endogenous growth models it also afects negatively the long-run rate of growth. In this paper, we explore the analytical relationship between the determinants of both depreciation and growth, in the context of the one sector model of growth when a linear production technology is combined with adjustment costs and a technology for capital maintenance. Agents are allowed to under-use the installed capital as well as to vary the depreciation rate. Therefore, in this economy agents decide endogenously the amount of resources to be devoted to the accumulation of new capital and also the amount to be devoted to repair and maintenance activities. The latter decision appears directly related to the matter of the endogeneity of both the capital utilization rate and the depreciation rate.

The issue of the maintenance of capital stock, which allows to break down the strong hypothesis of a constant depreciation rate even in the absence of obsolescence, has been left to one side for many years after the seminal contributions of the seventies. Nevertheless, in this period various attempts to reintroduce the variability of the depreciation rate have been implemented by means of the hypothesis Depreciation-in-Use. Namely, the causality argument which connects biunivocally high (low) rates of capital utilization, usually associated with high (low) levels of economic activity, with higher (lower) depreciation rates. This hypothesis has been incorporated to microeconomic studies at the firm level [Epstein and Denny (1980), Bischof and Kokkelenberg (1987), Motahar (1992)] as well as to macroeconomic studies related to both the neoclassical growth theory [Rumbos and Auernheimer (2001)] and the real business cycle theory [Burnside and Eichenbaum (1996)]. Although the depreciation rate is transformed into an endogenous variable, this approach does not seem to be completely satisfactory because of the residual role assigned to capital depreciation. More recently, the above hypothesis has been enlarged to include the maintenance activity, which allows for the depreciation rate to be a decision variable analogous to the capital utilization rate. We would like to mention the efort made at the firm level by Boucekkine and Ruiz-Tamarit (2003) as well as at an aggregate level in a neoclassical growth model by Licandro, Puch and Ruiz-Tamarit (2001) and in a real business cycle model by Licandro and Puch (2000) and Collard and Kollintzas (2000).

As mentioned above, the basic general equilibrium growth models do not allow for the separation of household saving decisions from investment decisions of firms. However, by introducing adjustment costs connected with gross investment expenditures it is possible to overcome the essentially passive role of investment in these models.1 In this paper, we take the canonical model of Rebelo (1991) and introduce both an adjustment cost function and a maintenance cost function which modify the standard objective functional. It is well-known that in the one-sector models of growth, the linear technology constitutes a useful referent for easily modeling, directly or asymptotically, the endogenous growth phenomenon. Consequently, we want to know whether the introduction of those new functions into the model breaks down the previous association. In such a context, depreciation is no longer a residual variable. Together with investment and the rate of capital utilization, it becomes one of the instruments used by economic agents in setting their optimal plans. In short, our query is whether incorporating the maintenance and repair expenditures into the aggregate model of economic activity, will substantially change what we know about the short-run convergence hypothesis as well as about the determinants of the long-run rate of growth and other endogenous variables. Our technological assumptions allow to expand the basic model in such a way that well-defined investment, depreciation, and utilization functions may be derived. However, there are no theoretical contributions that we know of, aimed at the study of all these topics together. This paper is devoted to this end and, consequently, it is primarily dedicated to investigating the short-run dynamics and the long-run balanced growth path.

1The active role of investment has been studied by Abel and Blanchard (1983) in a neoclassical Ramsey-like model, and also by Barro and Sala-i-Martín (1992) in an endogenous

The article is organized as follows. Section 2 describes the economy and introduces the assumptions featuring the diferent parts of the general equilibrium model. In section 3, we solve the intertemporal optimization problem and study the resulting dynamic system which governs the economy. Sections 4 and 5 are devoted to obtaining and interpreting results, connecting with the empirical literature which parallels the present work. Finally, section 6 summarizes and highlights the central aspects of the model.

2 The economy

Let us consider an economy populated with many identical infinitely-lived individuals, Nt. Population is assumed to grow at a constant and exogenously given rate . We normalize the initial population to unity and then we get . Moreover, it is assumed that people facing to an infinite planning horizon will discount the future at a positive constant rate . Individual preferences are represented by an instantaneous utility function , which is assumed increasing, twice diferentiable and strictly concave. This function only depends on the per capita consumption and it is assumed that Inada conditions are satisfied, lim and

growth model of the AK type.

Moreover, there are many identical firms producing a single good. For simplicity, we assume that each firm uses a linear technology of the type, the capital stock being the only relevant factor.2 We interpret capital in a broad sense, so that it includes physical capital as well as human capital, which usually comes embodied in workers. In this sense, human capital is considered a rival and excludable factor as physical capital is. Labour measured as the number of workers and independent of the index of human capital, which is considered as perfect substitute for physical capital, is not necessary for production. Therefore, total current output is a function of the efectively used capital , where is the variable proportion of installed capital that firm decides to use, and of the eficiency parameter A which represents a constant technological level. This latter parameter may also be read as the marginal productivity as well as the average productivity of the efectively used capital. Thus, given the constant returns inherent to a linear production function, we write in per capita terms as

\[y _ {t} = A k _ {t} u _ {t}.\tag{1}\]

Because of our interest in long-run endogenous growth paths, we leave to one side the hypothesis of exogenous technological progress. Then, it is possible to identify more easily the consequences of the assumed constant returns to capital for the rate of growth, the rate of capital utilization and the depreciation rate.

In this economy, the produced single-good may be allocated to consumption, to the accumulation of new capital or to preserving the inherited capital. While current consumption contributes directly to increase welfare, the other uses of output are connected with the increase of the capital stock, which allows for a greater consumption in the future. In this context, accumulation of new capital has not only to do with investment purchases but also with adjustment or installation activities. Moreover, preservation of the old capital has to do with maintenance and repair activities. Consequently, we have to introduce in our framework the two corresponding cost functions.

2Similar results could be derived under a more general production function with constant returns to scale if we introduce, following Romer (1986), the learning-by-investing device together with the knowledge spillovers assumption.

First, 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 installed capital stock, . Then , where is the rate of gross investment over capital, and is assumed non-negative, twice diferentiable, increasing and strictly convex for , with and lim . Consequently, per capita adjustment costs may be written as

Second, in order to preserve the inherited capital stock, we assume that period by period it is possible to reduce the depreciation associated with the deterioration which arises from equipment ageing and use,3 by means of the corresponding maintenance and repair activities. These activities entail specific maintenance costs which are internal to the firm and, by assumption, will be represented by a linearly homogeneous function , decreasing in total depreciation, , and increasing in efectively used capital. Redefining variables we get m , where is the endogenous rate of depreciation over capital stock and is the intensity of use of the installed capital stock. The function , the average maintenance cost, is assumed non-negative, twice diferentiable, convex and linearly homogeneous. Furthermore, we assume and for and with lim . The larger the utilization of capital the larger the cost of maintenance, and the larger the maintenance costs the smaller the depreciation rate of capital.4 The homogeneity assumption implies that Consequently, per capita maintenance costs may be written as

The aggregate resource constraint is where . In per capita terms the resource constraint is deter-

3Here we are refering strictly to physical wear and tear but, contrary to the standard procedure, we take this depreciation as an economic phenomenon because firms are assumed to optimally decide how much resources have to be allocated to maintenance. In this paper we ignore obsolescence as a source of depreciation. Factors usually causing obsolescence are left to one side because of the assumed perfect malleability of capital.
4An equivalent representation of the problem would correspond to the assumption that the depreciation rate is a function of both the utilization rate and the rate of maintenance cost to capital. This alternative view has been adopted by Boucekkine and Ruiz-Tamarit (2003) in a partial equilibrium context to develop the study of firm investment and depreciation decisions.

mined by these two equations

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

\[\stackrel {\bullet} {k _ {t}} = \left(i _ {t} - \delta_ {t} - n\right) k _ {t},\tag{3}\]

where denotes the time derivative of per capita capital considered in its broad sense.

3 The optimization problem

In an economy without externalities and no other market failures such as imperfections or incompleteness, which could appear in conflict with the assumptions of any of the two basic welfare theorems, the competitive equilibrium solution to the intertemporal resources allocation problem will correspond to the central planner solution. Therefore, in our model every optimal solution may be decentralized as a competitive equilibrium. The planner’s optimization problem is to choose at each moment in time the three controls: the rate of capital utilization, the rate of investment and the rate of depreciation, which solve the problem

\[\max _ {\{u _ {t}, i _ {t}, \delta_ {t} \}} W = \int_ {0} ^ {\infty} U (c _ {t}) e ^ {- (\rho - n) t} d t \quad \text {s. t. (2), (3) and k_{0}}.\tag{P}\]

The current value Hamiltonian associated with this problem, after dropping time subscripts, may be written as

\[H ^ {c} = U (A k u - [ i + \phi (i) + m (\delta , u) ] k) + \mu [ i - \delta - n ] k,\]

where is a co-state variable. According to the Maximum Principle,5 an interior optimal solution to problem (P) must satisfy the first-order conditions

\[A = m _ {u} (\delta , u),\tag{4}\]

\[\mu = U ^ {\prime} (c) [ 1 + \phi^ {\prime} (i) ],\tag{5}\]

5Under the more restrictive assumption that the Hamiltonian function is strictly concave with respect to the control variables, the solution functions are continuous and the first order conditions become necessary and suficient for a maximum.

\[\mu = - U ^ {\prime} (c) m _ {\delta} (\delta , u),\tag{6}\]

the Euler equation

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

the constraints (2) and (3), as well as the initial condition and the corresponding transversality condition

\[\lim _ {t \to \infty} e ^ {- (\rho - n) t} \mu_ {t} k _ {t} = 0.\tag{8}\]

The multiplier defines the shadow price, measured in units of utility, of an additional unit of installed capital. The term is the marginal opportunity cost of gross investment. Then, equation (5) states that this marginal cost measured in units of utility must be equal to the shadow price of capital. On the other hand, is the marginal saving in maintenance costs associated with an increase in the depreciation rate. An increase in reduces capital stock and, consequently, diminishes maintenance expenditures. So, equation (6) states that this marginal saving measured in units of utility must be equal to the shadow price of the lost capital. The term is the marginal maintenance cost associated with an increase in the utilization rate. Equation (4) states that this marginal cost must be equal to the marginal productivity of such an increase in the utilization rate, measured by the term A. Two additional relationships are fundamental for subsequent analysis

\[- \frac {m _ {\delta \delta} (\delta , u)}{m _ {\delta u} (\delta , u)} = - \frac {m _ {\delta u} (\delta , u)}{m _ {u u} (\delta , u)} = \frac {u}{\delta},\tag{9}\]

\[1 + \phi^ {\prime} (i) = - m _ {\delta} (\delta , u).\tag{10}\]

The first one represents the linear homogeneity assumption on the maintenance cost function. The second one, given that the planner has two alternative ways for increasing capital: investment and maintenance, states that at the optimum the marginal cost of investing has to be equal to the marginal cost of reducing depreciation through maintenance.

Consider now equation (7) and, after some manipulations, solve forward subject to the transversality condition (8) which avoids explosive solutions. Then, the price appears determined as the present discounted value of the total marginal product of capital measured in units of utility,

\[\mu_ {t} = \int_ {t} ^ {\infty} U ^ {\prime} (c _ {s}) [ A u _ {s} + i _ {s} \phi^ {\prime} (i _ {s}) - \phi (i _ {s}) - m (\delta_ {s}, u _ {s}) ] e ^ {- \int_ {t} ^ {s} [ \rho + \delta_ {z} ] d z} d s.\tag{11}\]

In this expression, the discount term takes into account the fact that the depreciation rate is variable.

Finally, consider the first order conditions (4)-(6), the resource constraint (2) and the production function (1), which implicitly define the optimal functions relating each control variable to the state and co-state variables. These control functions may be represented as ， and , where Θ represents a vector of structural parameters. All these functions are analyzed in Appendix I.

4 The dynamic system

We study now the dynamic system which describes the evolution of state and costate variables. First, we introduce the control functions and transform the accumulation equation (3) into the diferential equation

\[\stackrel {\bullet} {k} = [ i - \delta (k, \mu) - n ] k.\tag{12}\]

Then, making use of (9), (10) and (4)-(6), which allow for further simplifications as shown in Appendix II, the Euler equation (7) may be written as

\[\dot {\mu} = [ - H (i) + \rho ] \mu .\tag{13}\]

In this equation, the coeficient on the right hand side involves the function , which represents the whole marginal efects of investment on the Hamiltonian function; that is the full marginal productivity of investment or the social cost of transferring resources to the future. By the assumed convexity on , it takes only positive values for any and is a monotonous increasing function because of and

We have shown in Appendix I the constancy of the investment rate at the optimum. Therefore, the function gives a constant value and, hence, we get a constant coeficient in (13). Consequently, the system (12)-(13) has a structure very similar to the one characterizing standard AK models. However, we cannot go beyond because of two reasons. First, we cannot translate the above system from the state-costate space to the state-control space since our first order conditions do not allow for a simple substitution as in the basic model, where there is only one control variable and the transformed dynamic system becomes linear. Second, the depreciation rate appearing in (12) is a generic control function and we cannot identify any partial and separated linear form with respect to the state variable. Consequently, if we want to solve analytically this non-linear system, we need to consider particular forms for each of the structural function in the model. These ones, as were considered in Appendix I, are and m Then, given that in such a case , we can write our particular dynamic system as

\[\stackrel {\bullet} {k} = \left[ \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} - n \right] k - \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {\frac {1}{\Phi} - 1} \mu^ {\frac {- 1}{\Phi}},\tag{14}\]

\[\stackrel {\bullet} {\mu} = \left[ - \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \rho \right] \mu .\tag{15}\]

This system is non-linear and it does not admit a linearization because of the lack of a well defined steady state. However, its structure allows for a complete closed form solution.6

5 Closed-form solution and economic results

The unique non-explosive particular solution trajectories for the variables of the system are

\[k (t, \Theta) = k _ {0} \mathrm{exp} \left\{\frac {1}{\Phi} \left[ \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} - \rho \right] t \right\},\tag{16}\]

\[\mu (t, \Theta) = \mu (0) \exp \left\{\left[ \rho - \frac {\left[ \frac {\varepsilon}{d _ {\varepsilon} ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d _ {\varepsilon} ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} \right] t \right\},\tag{17}\]

6The method used to solve in closed form this non-linear modified Hamiltonian dynamic system may be found in Appendix III.

with known and

\[\mu \left(0\right) = \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {1 - \Phi}}{\left[ \frac {\left(1 - \frac {1}{\Phi}\right) \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \frac {\rho}{\Phi} - n \right] ^ {\Phi}} \frac {1}{k _ {0} ^ {\Phi}}.\tag{18}\]

Then, using (16) and (17) we compute the term which is needed in order to determine the complete particular trajectories for control variables. We find , and substituting in equations (I.1)-(I.3) from Appendix I we get the explicit trajectories for i(t, Θ), δ(t, Θ) and ,

\[i (\Theta) = \frac {1}{b} \left[ \frac {\varepsilon}{d _ {\varepsilon} ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right],\tag{19}\]

\[\begin{array}{l} \delta (\Theta) = \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {2} - 1 + \left(1 - \frac {1}{\Phi}\right) \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \frac {\rho}{\Phi} - n, \\ (\Theta) = \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {2} - 1 + \left(1 - \frac {1}{\Phi}\right) \left[ \frac {\varepsilon}{d ^ {\underline {{1}}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{2 b \varepsilon \left(\frac {A}{1 + \varepsilon}\right)} + \frac {\left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1}{\varepsilon}} \left(\frac {\rho}{\Phi} - n\right)}{d ^ {\frac {1}{\varepsilon}}}. \end{array} \tag {20}\tag{21}\]

The investment rate, the depreciation rate as well as the capital utilization rate are constant along the particular solution trajectory. The result concerning the investment rate could have been anticipated because of the previously proved independence of its associated control function with respect to the endogenous variables. However, in the case of the depreciation and utilization rates, the aforesaid result is due to the compensating efects exerted by capital stock and its shadow price on each of the variables along the solution trajectory.

Moreover, from definitions (1) and (2) and the above trajectories for controls, we derive the particular solution trajectories for output and consumption per capita

\[y (t, \Theta) = A u (\Theta) k _ {0} \mathrm{exp} \left\{\frac {1}{\Phi} \left[ \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} - \rho \right] t \right\},\tag{22}\]

\[c (t, \Theta) = \Gamma (\Theta) k _ {0} \mathrm{exp} \left\{\frac {1}{\Phi} \left[ \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} - \rho \right] t \right\}.\tag{23}\]

Here is time independent and represents the ratio , which is constant along the non-explosive solution trajectory,

\[\Gamma (\Theta) = \left[ \frac {\varepsilon}{d _ {\varepsilon} ^ {1}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] \left[ \frac {(1 - \frac {1}{\Phi}) \left[ \frac {\varepsilon}{d _ {\varepsilon} ^ {1}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d _ {\varepsilon} ^ {1}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \frac {\rho}{\Phi} - n \right].\tag{24}\]

In fact, the previous results characterize a balanced growth path. Namely, the model does not show transitional dynamics. Given the complete closedform solution for each of the involved variables, it is easy to conclude about the growth rates

\[\gamma_ {i} = \gamma_ {\delta} = \gamma_ {u} = 0,\tag{25}\]

\[\gamma_ {k} = \gamma_ {y} = \gamma_ {c} = \gamma = \frac {1}{\Phi} [ H (i) - \rho ] = \frac {1}{\Phi} \left[ \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} - \rho \right],\tag{26}\]

which is positive for . The growth rate does not depend on the initial capital stock and is not related to the initial income level or to any other per capita income level. The absence of convergence may be illustrated by taking two equally parameterized economies except for their initial capital stocks. In such a case, any initial per capita income diference between the two economies will always be increased over time, never reduced. In relative terms, if we take the ratio we observe that it remains constant equal to Only diferences in the structural parameters that make could modify the value of such a ratio, but the absence of transition leaves the dynamic conclusion unchanged. Therefore, the new elements considered in this model do not change the convergence conclusions deduced from the canonical AK model. The model accounts for most of the growth facts that are reported in Parente and Prescott (1993). It can explain the great disparity between rich and poor countries and the constancy of the disparity over time, even in a world where all countries including the poorest ones become somewhat richer.7 However, it cannot explain the demonstrated ability of some countries to change their positions within the per capita income distribution, i.e. miracles and disasters associated with overtaking processes.

In our model the saving rate, defined as , takes the constant value

\[s (\Theta) = \frac {A u (\Theta) - \Gamma (\Theta)}{A u (\Theta)} = \frac {i (\Theta) + \phi (i (\Theta)) + m (\delta (\Theta) , u (\Theta))}{A u (\Theta)}.\tag{27}\]

Household saving just finances the two kind of expenditures related to capital accumulation process: gross investment expenditures, including adjustment costs, and capital maintenance expenditures.

Moreover, given the absence of transitional dynamics, along the balanced growth path welfare depends on the initial consumption level and the constant rate of growth. If we substitute for the known values we get

\[\begin{array}{c} W (\Theta) = \int_ {0} ^ {\infty} \frac {c (t , \Theta) ^ {1 - \Phi} - 1}{1 - \Phi} e ^ {- (\rho - n) t} d t \\ = \frac {1}{1 - \Phi} \left(\frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {1 - \Phi} k _ {0} ^ {1 - \Phi}}{\left[ \frac {(1 - \frac {1}{\Phi}) \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right] ^ {2}}{\frac {2 b \varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \frac {\rho}{\Phi} - n} \right] ^ {\Phi} - \frac {1}{\rho - n}\right). \end{array}\tag{28}\]

Although it is not easy to compute their individual impacts, this expression shows the optimal welfare as a function depending only on the structural parameters of the model.

Finally, in the following proposition we supply an additional result which is interesting from the point of view of the empirical literature.

Proposition 1 Under the assumed functional forms for utility and adjustment and maintenance costs, then the rate of depreciation δ, the rate of capital utilization u, the investment rate i and the rate of growth γ are positively related.

7Easterly and Levine (2000), however, finds in data a massive divergence in the absolute levels of income per capita over the last thirty years, which implies that the rich grew faster than the poor.

Proof. Given that (4), under the particular functional forms, can be written as 4

\[\left(\frac {u}{\delta}\right) ^ {\varepsilon} = \frac {A}{(1 + \varepsilon) d} > 0,\tag{4'}\]

we conclude that u and are positively related.

Moreover, given that (10), under the particular functional forms, can be written as

\[i = \frac {\varepsilon d}{b} \left(\frac {u}{\delta}\right) ^ {1 + \varepsilon} - \frac {1}{b},\tag{10'}\]

we also conclude that i and u are positively related.

According to (26), given the parameters of preferences and and the particular functional forms, there is a positive relationship between and i because

Finally, from (3) and (26) we get

\[\delta = i (\gamma) - \gamma - n,\tag{3'}\]

which given n allows us to deduce a positive relationship between and , under the condition , i.e. . Such a condition is not too strong given that 1 and, from empirical studies, we know that . This relationship closes the loop and ends the proof. ¥

Therefore, all the endogenous variables will move in the same direction, allowing us to identify a strong positive correlation among them. According to the traditional AK model, the growth rate appears positively related to investment but negatively to the depreciation rate. In our model, however, the growth rate appears positively related to the investment rate but also to the depreciation and capital utilization rates. Consequently, a country with higher (lower) depreciation and utilization rates will also experience a higher (lower) rate of growth.

5.1 Comparative statics

In this section we show the results arising from a complete comparative statics exercise for variables i, δ, u and . Most of them cannot be found in the literature because of the insuficiencies of the canonical endogenous growth model, in which the depreciation rate is assumed constant and the capital stock is used at full capacity. From (19), (20), (21) and (26) we find that

1. The greater the productivity of efectively used capital or eficiency parameter A, the higher the investment rate as well as the depreciation and capital utilization rates.

2. The higher the weight of installation and maintenance costs in gross product, represented by parameters b and d respectively, the lower the investment rate as well as the depreciation and capital utilization rates.

3. Both the capital utilization rate and the depreciation rate are positively related with the level of impatience characterizing economic agents, which is represented by a low intertemporal elasticity of substitution in consumption, , and/or a high rate of discount, ρ.

4. The bigger the rate of population growth the lower the depreciation and capital utilization rates.

5. The investment rate does not depend on preference parameters or on the rate of population growth.

6. The higher the weight of installation and maintenance costs in gross product, b and d respectively, the lower the rate of growth.

7. The greater the eficiency level of efectively used capital A the higher the rate of growth.

8. The greater the patience of economic agents, i.e. the higher and/or the lower the higher the rate of growth.

9. The economy’s growth rate does not depend on the rate of population growth.

The new results about depreciation and growth that we have derived along the paper, including some of the previous comparative statics results, are consistent with empirical facts as recently have been reported by Gylfason and Zoega (2001). However, the results concerning the relationship between capital utilization and growth are still to be empirically checked.

6 Conclusions

As in standard AK models, our model does not show transitional dynamics. Variables k, y and c conform a unique balanced growth path from the beginning, while variables i, δ and u stand at their initial constant values forever. In addition, associated with the constant investment rate we get a constant saving rate and a constant consumption-capital ratio. This is so because the adjustment and maintenance cost functions included here are assumed linear with respect to k, but it could be diferent if we break down such an assumption.

Among the new results obtained in this paper some parameter dependences must be highlighted. We find that the smaller the installation and maintenance costs and the greater the economy’s eficiency level, the higher the investment, utilization and depreciation rates as well as the rate of growth. Moreover, the higher the rate of population growth, the lower the depreciation and capital utilization rates, although this parameter does not afect the investment rate or the rate of growth. Finally, the greater the patience level of economic agents, the higher the rate of growth and the lower the depreciation and utilization rates.

Looking at the relationship between the endogenous variables of the model, we identify a strong positive correlation between the rate of growth and the investment, utilization and depreciation rates. According to the traditional AK model, the endogenous rate of growth appears positively related to investment and negatively to the depreciation rate, while the capital stock is assumed to be fully used. In our model, the rate of growth is positively related to the investment rate but also to the depreciation and capital utilization rates. This result is in accordance to the observed facts concerning growth and depreciation but contradicts some previously accepted theoretical results.

Furthermore, because of the direct influence of the utilization rate, a lower capital being used more intensively may produce more output than otherwise. It could be perfectly possible that the economy with a lower capital stock produces and consumes more than the one with a higher capital stock. However, a greater per capita consumption will usually appear associated with a greater per capita production. In any case, the absence of convergence implies the amplification of any initial diference. Hence, our model may explain the great disparity between rich and poor countries as well as the persistence of such a disparity over time, but we cannot explain the experiences of growth which are known as miracles.

7 Appendix I

From and (1) we define implicitly the control functions: and where Θ represents the vector of structural parameters. By total diferentiation, the implicit function theorem allows us to identify the partial efects

\[\begin{array}{r l r} {u _ {k} = \frac {- m _ {\delta u}}{m _ {u u}} \frac {A u - i - \phi - m}{m _ {\delta} k} < 0,} & & {u _ {\mu} = \frac {- m _ {\delta u}}{m _ {u u}} \frac {1}{(m _ {\delta}) ^ {2} k U ^ {\prime \prime}} < 0,} \\ {i _ {k} = 0,} & & {i _ {\mu} = 0,} \\ {\delta_ {k} = \frac {A u - i - \phi - m}{m _ {\delta} k} < 0,} & & {\delta_ {\mu} = \frac {1}{(m _ {\delta}) ^ {2} k U ^ {\prime \prime}} < 0,} \\ {c _ {k} = 0,} & & {c _ {\mu} = \frac {- 1}{m _ {\delta} U ^ {\prime \prime}} < 0,} \\ {y _ {k} = \frac {A m _ {\delta u} (i + \phi)}{m _ {\delta} m _ {u u}} > 0,} & & {y _ {\mu} = \frac {- A m _ {\delta u}}{m _ {u u} (m _ {\delta}) ^ {2} U ^ {\prime \prime}} < 0.} \end{array}\]

These results show some interesting features of the model. First, given that capital stock and its shadow price evolve in opposite directions, it is very dificult to decide at first sight the evolution of the variables capital utilization rate and depreciation rate. Second, the investment rate remains constant for a given set of parameter values, implying that the gross investment share will move in parallel to the capital-output ratio. Finally, per capita consumption evolves inversely to the movement in the shadow price of capital stock, as it happens with production per capita which, in addition, moves directly with capital stock.

Now, we illustrate the previous statements about control functions by specifying particular forms for each structural function of the model. We consider a CRRA instantaneous utility function , where Φ is a non negative constant representing the inverse of the intertemporal elasticity of substitution. Per capita production is obtained from a linear technology depending on the efectively used capital stock . Adjustment costs are represented by , where b is a positive constant. Maintenance costs are represented by , where approaches the elasticity of average maintenance cost with respect to the depreciation rate and the utilization rate, and d is a positive constant. These particular functions satisfy all the assumed general properties.

Solving the optimization problem and focussing on the control functions we get

\[i (\Theta) = \frac {1}{b} \left[ \frac {\varepsilon}{d _ {\varepsilon} ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} - 1 \right],\tag{I.1}\]

\[\delta (k, \mu , \Theta) = \frac {\frac {1}{2 b} \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {2} - \frac {1}{2 b}}{\frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}}} + \frac {\mu^ {\frac {- 1}{\Phi}} k ^ {- 1}}{\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {1 - \frac {1}{\Phi}}},\tag{I.2}\]

\[u (k, \mu , \Theta) = \frac {\frac {1}{2 b} \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {2} - \frac {1}{2 b}}{\varepsilon \left(\frac {A}{1 + \varepsilon}\right)} + \frac {\left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1}{\varepsilon}} \mu^ {\frac {- 1}{\Phi}} k ^ {- 1}}{d ^ {\frac {1}{\varepsilon}} \left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {1 - \frac {1}{\Phi}}}.\tag{I.3}\]

The expressions for consumption and output may be derived substituting the previous ones in the resources constraint (2) and the production function (1). As we can see, investment only depends on structural parameters and, hence, it takes a constant value as in equation (I.1). Equations (I.2) and (I.3) show expressions for δ and , which depend negatively on the state and costate variables. Moreover, these two variables are linearly and positively related to each other.

8 Appendix II

Consider the first order conditions (4)-(6) and the Euler equation (7) written as

\[\bullet \mu = \frac {\mu \left\{- f ^ {\prime} (k u) u + \phi (i) - i \phi^ {\prime} (i) + m (\delta , u) + [ 1 + \phi^ {\prime} (i) ] (\delta + \rho) \right\}}{[ 1 + \phi^ {\prime} (i) ]},\tag{II.1}\]

where because of the AK nature of the model. Now, define the function , which by the assumed convexity on gives positive values for any . This function is a monotonous increasing function because of and . Moreover, we know that for any . Then, equation (II.1) may be rewritten as

\[\stackrel {\bullet} {\mu} = \mu \left[ \rho + \delta + \frac {m (\delta , u)}{1 + \phi^ {\prime} (i)} - H (i) - \frac {f ^ {\prime} (k u) u}{1 + \phi^ {\prime} (i)} \right].\tag{II.2}\]

In this model, by analogy to the standard models, the term is the total gross marginal product of capital. Hence, we define the total net marginal product of capital as

\[r = \frac {f ^ {\prime} (k u) u}{1 + \phi^ {\prime} (i)} + H (i) - \frac {m (\delta , u)}{1 + \phi^ {\prime} (i)} - \delta .\tag{II.3}\]

Given the linear homogeneity assumption on the maintenance cost function, the above expression reduces to . Consequently, it represents the social cost of transferring resources to the future, i.e. the interest rate at the decentralized equilibrium. The Euler equation may be simplified to

\[\dot {\mu} = - \mu [ r - \rho ] = \mu [ - H (i) + \rho ],\tag{II.4}\]

which corresponds to equation (13) in the main text. Moreover, given the definition of H (i), we know that it express in a summarized way the whole marginal efects of investment on the Hamiltonian function, that is the full marginal productivity of investment. Implicitly, this function includes both the higher investment expenditures and the lower adjustment costs due to an increase in capital stock.

An alternative but complementary view of that function may be obtained from a simple reorganization of terms which leads to

\[H (i) = U ^ {\prime} (c) \frac {i}{\mu} [ \phi^ {\prime} (i) - \frac {\phi (i)}{i} ].\tag{II.5}\]

The term on the right hand side shows the diference between marginal and average adjustment cost multiplied by the investment rate and divided by the shadow price of capital. This value is transformed into utility units after we multiply by the marginal utility of consumption.

9 Appendix III

The dynamic system (14)-(15) may be solved using sequentially the two diferential equations together with the boundary conditions. However, to be exhaustive in our search of a closed-form solution and study the issues of existence, uniqueness and positivity along with transitional dynamics and long-run growth, we will take as reference the modified Hamiltonian dynamic system studied in Ruiz-Tamarit and Ventura-Marco (2000), which is of the form

\[\stackrel {\bullet} {k} (t) = \Delta_ {k} k (t) - \Omega_ {k} k (t) ^ {a _ {1 1}} \mu (t) ^ {a _ {2 2}},\tag{III.1}\]

\[\stackrel {\bullet} {\mu} (t) = \Delta_ {\mu} \mu (t) + \Omega_ {\mu} k (t) ^ {a _ {1 1} - 1} \mu (t) ^ {1 + a _ {2 2}},\tag{III.2}\]

\[k (t _ {0}) = k _ {0},\tag{III.3}\]

\[\lim _ {t \to \infty} \mu (t) k (t) \exp \left\{- (\rho - n) (t - t _ {0}) \right\} = 0.\tag{III.4}\]

The elements and are constant parameters, while and t are the variables. Moreover, the next general parameter constraints are assumed: and

It is easy to show that the above dynamic system simplifies to (14)-(15) under the following specific parameter values: and . From these values we can see that , where the right hand term represents the efective intertemporal rate of discount. At the same time, we have provided that or, according to what we saw in Appendix II, as long as the net marginal product of capital r is always higher than the rate of time preference This also implies as well as . Under these conditions we proceed in three steps. First, define the instrumental variable . By totally diferentiating and substituting from equations (III.1) and (III.2) we get

\[\stackrel {\bullet} {X} (t) = a _ {x} X (t) - b _ {x}.\tag{III.5}\]

This is an autonomous non-homogeneous linear diferential equation with constant coeficients and 0. Given the initial condition and a certain, for the moment unknown, initial value which allow us to determine the initial condition , any particular solution to (III.5) must be of the form

\[X (t) = \frac {b _ {x}}{a _ {x}} + \left[ X (t _ {0}) - \frac {b _ {x}}{a _ {x}} \right] \exp \left\{a _ {x} (t - t _ {0}) \right\}.\tag{III.6}\]

Once we know the given value of every parameter and the initial ones of the variables, the above expression determines the value of the instrumental variable at any moment in time. In a second step we transform the initial non-linear system and get the two separated, non-autonomous but homogeneous, linear diferential equations for the primary variables

\[\stackrel {\bullet} {k} (t) = \left(\Delta_ {k} - \frac {\Omega_ {k}}{X (t)}\right) k (t),\tag{III.7}\]

\[\stackrel {\bullet} {\mu} (t) = \Delta_ {\mu} \mu (t).\tag{III.8}\]

The expressions for the particular solutions are, respectively,

\[k (t) = k _ {0} \exp \left\{\int_ {t _ {0}} ^ {t} \left(H (i) - n - \frac {\left[ \frac {\varepsilon}{d ^ {\frac {1}{\varepsilon}}} \left(\frac {A}{1 + \varepsilon}\right) ^ {\frac {1 + \varepsilon}{\varepsilon}} \right] ^ {\frac {1}{\Phi} - 1}}{X (s)}\right) d s \right\},\tag{III.9}\]

\[\mu (t) = \mu (t _ {0}) \exp \left\{- (H (i) - \rho) (t - t _ {0}) \right\}.\tag{III.10}\]

The third step consists in determining the initial value of the costate variable for which trajectories are non-explosive. Given known, this may be done by determining . All what is needed in this step can be deduced from the transversality condition. This necessary condition, given the signs of the parameters, may be simplified to

\[\lim _ {t \to \infty} \left| \frac {b _ {x} \exp \left\{- a _ {x} (t - t _ {0}) \right\}}{a _ {x} X (t _ {0})} + 1 - \frac {b _ {x}}{a _ {x} X (t _ {0})} \right| = 0.\tag{III.11}\]

In particular, given that , this condition holds if, and only if, both and hold. Coming back to (III.6) we find that the instrumental variable will remain constant and equal to its initial stationary value Consequently, the non-explosive solution trajectories for variables involved in the modified Hamiltonian dynamic system (III.1)-(III.4) are unique and may be written as in equations (16), (17) and (18). Finally, the constraint also holds. Therefore, Proposition 4 from Ruiz-Tamarit and Ventura-Marco (2000) applies here and we conclude that

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DOCUMENTOS DE TRABAJO

References

  1. 2004-21: “Endogenous Growth, Capital Utilization and Depreciation”, J. Aznar-Márquez y J. R. Ruiz-Tamarit.

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  1. 2004-18: “Demographic change, immigration, and the labour market: A European perspective”, Juan F. Jimeno.

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  1. 2004-15: “Non-Catastrophic Endogenous Growth and the Environmental Kuznets Curve”, J. Aznar-Márquez y J. R. Ruiz-Tamarit.

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  1. 2204-14: “Proyecciones del sistema educativo español ante el boom inmigratorio”, Javier Alonso y Simón Sosvilla-Rivero.

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  1. 2004-09: “Could this ever happen in Spain? Economic and policy aspects of a SARS-like episode”, José A. Herce.

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  1. 2004-08: “Capital humano en España: Una estimación del nivel de estudios alcanzado”, Javier Alonso y Simón Sosvilla-Rivero.

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  1. 2004-04: “Currency Crises and Political Factors: Drawing Lessons from the EMS Experience”, Francisco Pérez-Bermejo y Simón Sosvilla-Rivero

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  1. 2004-03: “El futuro de las pensiones en España: Perspectivas y lecciones”, J. Ignacio Conde-Ruiz y Javier Alonso.

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  1. 2004-02: “Do temporary contracts increase work accidents? A microeconometric comparison between Italy and Spain”, Virginia Hernanz y Luis Toharia.

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  1. 2004-01: “Job Match Quality throughout the Business Cycle in the Spanish Labour Market”, Cristina Fernández.

TEXTOS EXPRESS

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  1. 2004-01: “The Spanish economy through the recent slowdown. Current situation and issues for the immediate future”, José A. Herce y Juan F. Jimeno.