Numerical solution by iterative methods of a class of vintage capital models* by Raouf Boucekkine**, Marc Germain*** Omar Licandro**** and Alphonse Magnus***** DOCUMENTO DE TRABAJO 98-20

December, 1998

* We are extremely grateful to Ken Judd and to Luis Puch for their most stimulating suggestions. The numerical computations have been performed on the Convex Exemplar SPP-1600 of the Université Catholique de Louvain.

** UCI-RESL.

*** CORE.

**** FEDEA.

***** UCL.

Los Documentos de trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es/hojas/publicaciones.html#Documentos de Trabajo

These Working Documents are distributed free of charge to University Department and other Research Centres. They are also available through Internet: http://www.fedea.es/hojas/publicaciones.html#Documentos de Trabajo

Raouf Boucekkine UCL-IRES

Marc Germain CORE

Omar Licandro FEDEA

Alphonse Magnus UCL

December, 1998

Abstract

We build up an iterative numerical procedure in order to solve vintage capital growth models with nonlinear utility functions and Leontieff technologies, a class of models intensively used in the literature since the early 90's. The numerical procedure is of the relaxation type and uses a step by step maximization scheme for updating. The procedure is close to the cyclic coordinate descent algorithm as described in the computational mathematics literature. We explain why and how our numerical scheme is suitable to handle the considered class of models.

Keywords: Vintage capital models, Relaxation, Optimization, Cyclic coordinate descent algorithm.

JEL classification: C63; E32; O40

*We are extremely grateful to Ken Judd and to Luis Puch for their most stimulating suggestions. The numerical computations have been performed on the Convex Exemplar SPP-1600 of the Université Catholique de Louvain. Correspondence: Raouf Boucekkine, Université Catholique de Louvain, IRES, Place Montesquieu, 3, B-1348 Louvain-la-Neuve, Belgium.

1 Introduction

The analysis of vintage capital growth models has regained interest since the early 90's. Indeed, this kind of models allow to address quite conveniently many of the current key economic issues like investment volatility, equipment replacement and the general consequences of the schumpeterian creative destruction process. Investment volatility and equipment replacement are analyzed using vintage capital models by Benhabib and Rustichini (1993) and by Boucekkine et al. (1997a, 1998). Using the same framework, Aghion and Howitt (1994) and Caballero and Hammour (1996), among others, have analyzed the effects of creative destruction on unemployment and job reallocation. Except in Benhabib-Rustichini's paper in which a general CES production function is adopted, all the previous contributions use additionally a Leontieff technology. That is this type of technologies with complementary production factors straightforwardly ensures an endogenous determination of the equipment replacement decision, while gross substitutability may directly induce infinite optimal lifetimes for equipment, which sounds unrealistic. Moreover, when the utility functions are linear, Leontieff technologies allow to bring out some analytical results as in Boucekkine et al. (1997a).

When the utility functions are non-linear, numerical resolution is unavoidable. The major difficulty in handling numerically optimal growth vintage capital models with the latter feature and with Leontieff technologies, results in the treatment of the endogenous leads appearing in the associated optimality necessary conditions. Precisely, these conditions together with Leontieff specifications yield systems of differential-difference or integro-differential-difference equations with state dependent lags and leads. As long as only state dependent lags or only state dependent leads are involved, the resulting systems can be solved using, directly or with some adaptation work, some well-known algorithms (see Baker and Paul, 1996, and Boucekkine et al., 1997b). However, optimal vintage capital growth models do involve simultaneously state dependent leads and lags. So far, no robust numerical solver for such systems has been proposed. In the economic literature, a worthwhile attempt at solving a system of this type is due to Caballero and

Creative destruction reflects the idea that economic growth is mainly guided by successive innovations leading to the “death” of the resulting less productive techniques and equipment.

Hammour (1996). However, the treatment of these authors cannot be applied to a general system of equations with endogenous leads and lags since it is based on some very particular assumptions. Especially, having introduced a periodic forcing variable in their model, the authors assume that the solution paths are also periodic and succeed at locating them using some standard algorithms in the literature of two-point boundary value systems.

As the latter treatment seems at best only implementable on differential-difference systems driven by periodic forcing functions, it is not general. One way to overcome the problem arising from the simultaneous occurrence of state dependent leads and lags consists in tackling the optimization problem directly without using the optimality conditions (where precisely the endogenous leads appear). The optimization work can be performed step by step within a standard fixed-point relaxation algorithm. This strategy is applied by Boucekkine et al. (1998) for example. Indeed, this device can be applied to a large class of vintage models of the recent economic literature, especially those including a Leontieff technology (like those quoted above for example). For convenience, we prefer to focus on this class of models since the specifications have been extremely popular in this decade.

Indeed, as it will be clear later, our method is also suitable to handle vintage capital models with non Leontieff production functions, for example with putty-clay technologies as in Caballero and Hammour, 1998, who use an ex ante CES function. According to the interpretation of these authors, this kind of ex-ante production function can be thought of as an envelope of possible Leontieff functions whose technologies can be developed ex-post. Therefore, it is not surprising at all that the technical problems involved in such cases are almost identical to those we face when dealing with vintage capital models with Leontieff technology. As for the issues related to the numerical resolution of the latter models when preferences are nonlinear, our approach has three important advantages:

i) As explained just above, it allows to avoid the numerical difficulties coming from the simultaneous presence of endogenous leads and lags.

ii) As theoretically and numerically shown in Boucekkine et al (1997a, 1998), vintage capital models may give rise to corner and non-differentiable solutions. These features are obviously likely to be captured by an optimization algorithm based on the objective function. The relaxation algorithms so far used in the economic literature cannot do so because they usually handle the first order necessary conditions for interior solutions to exist.

iii) Last but not least, our optimization device can be seen as an application of the cyclic coordinate descent algorithm as described by Luenberger (1965), Section 7.8, for example. Depending on the model under consideration, numerous variations of this algorithm can be considered to improve the efficiency of the computational setting if necessary.

To illustrate the method, we apply it to the vintage capital model analyzed in Boucekkine et al. (1998). This model is presented in the next section. Section 3 presents the numerical setting and shows how it is related to the original optimization problem. Section 4 illustrates the working of the method through a very simple example. The concluding section 5 points at further potential improvements of the method.

2 Optimization Problem

Consider a centrally planned closed economy characterized by a clay-clay vintage capital technology, i.e., technical progress is embodied in new machines. These new machines are of the Leontieff type, and the scrapping of old machines is endogenous.

The central planner is supposed to maximize social welfare by solving the following problem:

\[\max W = \int_ {0} ^ {\infty} u [ c (t) ] \exp \{- \rho t \} d t\tag{1}\]

subject to

\[y (t) = \int_ {t - T (t)} ^ {t} i (z) \mathrm{d} z\tag{2}\]

\[\int_ {t - T (t)} ^ {t} i (z) \exp \{- \gamma z \} \mathrm{d} z = 1\tag{3}\]

\[y (t) = c (t) + i (t)\tag{4}\]

\[0 \leq i (t) \leq y (t)\]

given for all .

is consumption, is production, is investment and represents the age of the oldest operating machines, or scrapping time. The utility function is assumed to be , increasing and strictly concave. Parameter is strictly positive and represents the rate of time preference.

Machines from vintage t, , are supposed to produce one unit of output each, and do require units of labor. The parameter is positive and represents (Harrod neutral) technical progress. Total labor resources are assumed to be constant and equal to one. (2) is the Leontieff production function and (3) is the equilibrium condition on the labor market.

A necessary condition for an interior solution of this optimization problem to exist is:

\[u ^ {\prime} (c (t)) = \int_ {t} ^ {t + J (t)} \left(1 - \mathrm{e} ^ {\gamma (z - t - T (z))}\right) u ^ {\prime} (c (z)) \mathrm{e} ^ {- \rho (z - t)} \mathrm{d} z\tag{5}\]

with . is the expected lifetime of the equipment bought at t. The optimality condition above is the optimal (interior) investment rule: The marginal cost of investing, on the left hand side, must be equal to the marginal revenue, which depends on the future scrapping time of new machines. Indeed, this condition captures the forward-looking component of the model while (2) and (3) capture its backward-looking dimension. Differentiating all these equations with respect to time yields an integro-differential-difference system with an endogenous lag and an endogenous lead . No robust solver is so far available to handle this kind of systems. The next section presents an iterative method allowing to solve the considered optimization problem. The optimization work relies directly on the objective function, so it does not use explicitly the necessary conditions that cause the simultaneous presence of endogenous leads and lags to occur.

Before, let us briefly present the major dynamic property of this kind of models in order to allow the reader to understand the results of our experiments in section 4. Because new machines are more productive in our model, older machines are optimally scrapped to be replaced by the former. The resulting dynamics follow the so-called “echo principle”, i.e the ability of an economy to reproduce its own past history (see Benhabib and Rustichini, 1993, for a comprehensive analysis of this feature). In addition to that, the example model considered in this paper admits a balanced growth path. When the economy starts with a too high (Resp. low) past investment profile with respect to the balanced growth paths values, the adjustment to these balanced growth paths optimally induces an initial depression (Resp. expansion) of investment, which will be reproduced in the future according to the “echo principle”. These echo oscillations are indeed damped to allow convergence to the balanced growth paths.

All the properties of the model stated in this section are taken from Boucekkine et al. (1998).

3 Maximization by iteration: Numerical setting and theoretical justifications

3.1 The numerical setting

We decide to perform the approximate maximization of W, with by (4), by replacing the unknown functions i and y by piecewise constant functions on the intervals , , ..., Let , , ..., , , ... be the unknown values. Piecewise constant functions would be crude and not very satisfactory approximations to well-behaved smooth functions, but we want to be ready to cope with possibly discontinuous solutions, and prefer to make a robust and unsophisticated choice. Then, y-i and still are piecewise constant functions, the values of the latter one being , , ... Integrals of piecewise constant functions are computed exactly by the midpoint rule:

\[\int_ {0} ^ {(N + 1) \Delta} F (t) d t = \Delta \sum_ {k = 0} ^ {N} F ((k + 1 / 2) \Delta).\]

Remark that we keep a finite sum, even if we have to discretize an integral from 0 to , in which case a large value is chosen for N. If we have to perform an integral on an interval whose endpoints are not integer multiples of , we have to be careful with the first and the last terms:

\[\begin{array}{l l} \int_ {a} ^ {b} F (t) d t = (k _ {1} \Delta - a) F ((k _ {1} - 1 / 2) \Delta) & + \Delta F ((k _ {1} + 1 / 2) \Delta) + \dots + \Delta F ((k _ {2} - 1 / 2) \Delta) \\ & + (b - k _ {2} \Delta) F ((k _ {2} + 1 / 2) \Delta), \end{array}\]

where is the smallest integer multiple of which is larger or equal than a, and where is the largest integer multiple of which is smaller or equal than (see the figure below the discussion of the equation (7) for an example).

Finally, integrals of piecewise constant functions times exponential functions could be performed exactly, but we preferred to keep the midpoint rule, allowing for a small discrepancy, as is small, and the exponential functions are slowly varying. Hereafter we set , for , and we denote by and the values of investment and output at .

Application of these principles leads to the discrete-time problem analog to (1-4):

\[\max W _ {\mathrm{disc}} = \Delta \sum_ {k = 0} ^ {N} u (y _ {k} - i _ {k}) \exp (- \rho t _ {k}),\tag{6}\]

subject to

\[y _ {k} = (j \Delta - t _ {k} + T (t _ {k})) i _ {j - 1} + \Delta i _ {j} + \dots + \Delta i _ {k - 1} + \frac {\Delta}{2} i _ {k}, k = 0, 1, \ldots , N,\tag{7}\]

where is the smallest integer multiple of larger or equal than ,

\[1 = (j \Delta - t _ {k} + T (t _ {k})) i _ {j - 1} e ^ {- \gamma t _ {j - 1}} + \Delta i _ {j} e ^ {- \gamma t _ {j}} + \dots + \Delta i _ {k - 1} e ^ {- \gamma t _ {k - 1}} + \frac {\Delta}{2} i _ {k} e ^ {- \gamma t _ {k}}, k = 0, 1, \dots , N,\tag{8}\]

\[0 \leq i _ {k} \leq y _ {k}, k = 0, 1, \ldots , N.\]

given for all .

A word of explanation on how (1-4) is discretized in (6-8):

The computation of (6) must be possible as soon as we get a set of values of a trial function i, i.e., a set of numbers .

To this end, we need the corresponding numbers . These values of y are computed from (7) which is the exact value of the integral (2) at , as far as the function i is assumed to be piecewise constant. As the lower bound is normally not an integer multiple of , we have to consider the integer j such that . The figure makes it clear:

Well, everything is all right...provided is known. This one is now computed from (3), according to the same principles, assuming reasonably close to a piecewise constant function. The formula (8) is actually an equation for the unknown . We get the new area, advancing from right to left, summing the successive rectangular areas , up to the point where we get a number, say larger than one, by summing the last small rectangle . This means that we went too far to the left (that's how j is found) by an amount . The left endpoint of the integral is therefore not , but

\[t _ {k} - T (t _ {k}) = (j - 1) \Delta + (\alpha - 1) / [ i _ {j - 1} \exp (- \gamma t _ {j - 1}) ],\]

yielding the computed value of .

The maximization (6) is performed by iteration. At each iteration, we use a step by step maximization device. Precisely, at each step, we look at the influence of a single value associated to a time on the discretized integral , keeping unchanged the other investment ordinates. The whole algorithm can be described briefly as follows:

a) Initialize the investment vector , the base of the relaxation.

b) Maximization step by step

i) step 0 Maximize the discretized integral with respect to keeping unchanged all the subsequent investment ordinates with respect to the base. Update with the resulting maximand. Compute the resulting new values of and , from (7, 8).

ii) step k For , maximize the discretized integral with respect to keeping unchanged future investment ordinates , if any, with respect to the base, with the preceding investment ordinates , , updated thanks to the already performed maximization steps.

Update using the resulting maximand. Compute the resulting new values of and according to (7, 8).

c) Compare the relaxation base with the updated investment vector resulting from step b). If they are sufficiently close, stop the iterations. Otherwise, run another step b) with the updated investment vector as a base. The latter type of iteration should be repeated until locating a fixed-point investment vector.

As one can see, we follow basically a Gauss-Seidel approach for the location of the optimal solution paths. However, the implemented maximization device and the need to recompute y and T values at each step makes the whole algorithm much more complex than the standard Gauss-Seidel scheme. Indeed, we have right now still no satisfactory hints on the behaviour of the error through iterations c). It should be noted, that except the required (and costly) updating of y and T at each maximization step, our steps b) and c) correspond to the cyclic coordinate descent optimization algorithm as described in Luenberger (1965), pages 158-161, for unconstrained problems. This algorithm may be extremely appealing because of its easy implementation as it does not require any information on the gradient of the objective functions in contrast to most alternative optimization methods. It follows that the cyclic descent algorithm can prove an excellent resolution technique when optimization has to be performed with respect to a very large number of choice variables as gradient information is extremely costly in this case. Since we are optimizing with respect to an investment vector, , when N is large, the cyclic descent algorithm appears very attractive for our purposes. Although this algorithm is known to have slower convergence rates than certain gradient-based optimization methods, we find it most useful in practice. This is in part due to certain theoretical properties of the models under considerations as explained in the next subsection.

T,

Indeed, in our most refined tests, we took , about 1% of the average value of the scrapping time T, and a time solution horizon equal to 100, about 10 times the average T. So, we had about one-dimensional maximizations to perform per iteration. Finally, at least 100 iterations c) are needed to get a reasonable accuracy (around ).

c)

T.

As the popular steepest descent algorithm, see again Luenberger(1965), pages 148-154.

3.2 Theoretical justifications of the numerical setting

The following concavity proof validates the step by step optimization strategy followed in our numerical setting. Using this proof, we can also deduce a differential relation that shows clearly how and why the solutions provided by the algorithm are related to the optimality conditions of the original optimization problem.

Proposition. The discretized integral is a concave function, when considered as a function of a single ordinate , for any fixed .

Proof: Indeed, let be the value of in a situation where , where , a sufficiently small number and k is some nonnegative integer.

We now decide to give some new value on the interval of width and center , with unchanged in all the other intervals. The value of depends on all the changes induced in the various values . Let be another time abscissa. We have , so we have only to appreciate the change in , which depends on . From (8),

\[1 = \int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1}} i (z) _ {\beta} e ^ {- \gamma z} d z = \int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1}} i (z) _ {\alpha} e ^ {- \gamma z} d z + (\beta - \alpha) \Delta e ^ {- \gamma t _ {0}},\]

only if , as (8) will be kept unchanged in other cases. Let us subtract the “old” equation (8):

\[\begin{array}{r c l} {0} & {=} & {\int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1} - T (t _ {1}) _ {\alpha}} i (z) e ^ {- \gamma z} d z + (\beta - \alpha) \Delta e ^ {- \gamma t _ {0}}} \\ & {\approx} & {[ T (t _ {1}) _ {\beta} - T (t _ {1}) _ {\alpha} ] i (t _ {1} - T (t _ {1})) e ^ {- \gamma (t _ {1} - T (t _ {1})} + (\beta - \alpha) \Delta e ^ {- \gamma t _ {0}},} \end{array}\tag{9}\]

if is chosen to be close to , so that the change of is small.

We now look at the new value of :

\[\begin{array}{r} y (t _ {1}) _ {\beta} = \int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1}} i (z) _ {\beta} d z = \int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1}} i (z) _ {\alpha} d z + (\beta - \alpha) \Delta , \\ y (t _ {1}) _ {\beta} - y (t _ {1}) _ {\alpha} = \int_ {t _ {1} - T (t _ {1}) _ {\beta}} ^ {t _ {1} - T (t _ {1}) _ {\alpha}} i (z) _ {\alpha} d z + (\beta - \alpha) \Delta \approx [ T (t _ {1}) _ {\beta} - T (t _ {1}) _ {\alpha} ] i (t _ {1} - T (t _ {1})) + (\beta - \alpha) \Delta , \end{array}\]

Note that the sums appearing in (8) in Section 3.1 can be written as integrals as we are dealing with piecewise constant functions. This is done for convenience in this section.

which simplifies into

\[y (t _ {1}) _ {\beta} - y (t _ {1}) _ {\alpha} = (\beta - \alpha) \Delta \left[ 1 - \exp (\gamma (t _ {1} - T (t _ {1}) - t _ {0})) \right],\]

thanks to (9). We just got the increase of , as far as . At , the change is of course

\[c (t _ {0}) _ {\beta} - c (t _ {0}) _ {\alpha} = y (t _ {0}) - \beta - [ y (t _ {0}) - \alpha ] = \alpha - \beta ,\]

and the change for is

\[\begin{array}{r c l} W (\beta) - W (\alpha) & = & \Delta [ u (c (t _ {0}) _ {\alpha} + \alpha - \beta) - u (c (t _ {0}) _ {\alpha} ] e ^ {- \rho t _ {0}} \\ + \Delta \sum_ {t _ {1} - T (t _ {1}) \leq t _ {0} < t _ {1}} \Bigl \{u \Bigl (c (t _ {1}) _ {\alpha} + (\beta - \alpha) \Delta [ 1 - e ^ {\gamma (t _ {1} - T (t _ {1}) - t _ {0})} ] \Bigr) - u (c (t _ {1}) _ {\alpha}) \Bigr \} e ^ {- \rho t _ {1}}, \end{array}\tag{10}\]

for close to and for small .

As (10) is a linear combination of concave functions of , is indeed a concave function too, for in a small interval, but concavity must only be established locally to hold everywhere. ☐

As is now known to be a concave function of a single i ordinate, we are therefore sure that has a unique maximum in the admissible range, i.e., from x = 0 up to the value such that . An interesting consequence of (10) is the differential relation when :

\[W _ {d i s c} ^ {\prime} (x) = \Delta \left[ - u ^ {\prime} \left(c \left(t _ {0}\right)\right) e ^ {- \rho t _ {0}} + \int_ {t _ {0}} ^ {t _ {0} + J \left(t _ {0}\right)} u ^ {\prime} \left(c \left(t _ {1}\right)\right) \left[ 1 - e ^ {\gamma \left(t _ {1} - T \left(t _ {1}\right) - t _ {0}\right)} \right] e ^ {- \rho t _ {1}} d t _ {1} \right],\tag{11}\]

and for all yields the well known optimality condition (5), solving problem (1) with constraints (2-4). This shows how the numerical procedure is linked to this (forward-looking) optimality condition of the original optimization problem. Indeed, our numerical procedure performs basically as a Gauss-Seidel scheme applied to the (discretized) set of optimality conditions for an interior solution when it exists. When a corner solution emerges, our procedure also works since it is based on the objective function. Hence, our method is perfectly suitable to solve optimization-based models in which the optimal paths involve corner solutions, and it behaves as a Gauss-Seidel scheme applied to the first order conditions when an interior solution is encountered.

Assuming that all the differentiations are valid, or in other words that an interior solution exists for the optimization problem at .

4 An example

Let us solve the problem with , , , close to 1. So we have a weakly concave function to maximize, and we expect rather a rough behaviour of the solution paths. The balanced growth path solution is : if this law holds for t < 0, it will still hold when t > 0. As one expects a perturbation of this law, one makes the computations with approximating at , .

One decides to take on t < 0. This is much too high! At t = 0, the economy is allowed to react. What will happen?

Let us look at the record. One starts with trial values at t > 0, i.e., for all . Let us perform the approximate calculations with a (much too large) time step . We look at the discretized integral for several test values , keeping , , etc. unchanged:

The discretized integral is the largest when , which is not surprising, as was too large at t<0. We keep , and make move:

Again, is best. And we get the same answer for . Setting , we maximize with respect to :

Some change at last, the best appears to be close to 0.11. We know look at :

Here, the best value of is about 0.18. One goes on with next values (in order not to be bothered by truncated series effects in the estimation of the integral, we took a big upper bound, N = 500).

These successive maximizations of (step b) using the terminology of subsection 3.1) lead to the following first approximation to :

Most of the values stay close to the initial 0.1078. One sees slight depressions at 11 and 19, faint echoes of the big depression near the origin.

The job is not finished yet! We just performed maximization of on each , keeping unchanged the 's for j < k. This is only a limited way to see how interacts with its neighbours, and we start again a whole run of maximizations following exactly the same principles as before (step c) in the terminology of section 3.1).

Such a process is often called “relaxation”, as it was first used in the calculation of complicated mechanical devices, with lots of interacting parts. The calculation considered only a limited number of links, ignoring (i.e., relaxing) the other ones. Of course, when such a run of simplified calculations was achieved for all parts, there was still no exact matching, and the whole process was restarted again, and again... Given the concavity of our problem with respect to the investment variable, the amount of discrepancy between the solutions of two successive iterations tends to diminish along this process.

On our example, a second iteration (or a second run of maximizations) does not change the values of , and , which are still kept at the zero level. However, the best is found to be no more 0.11, but 0.04:

Things hardly changed:

After about 20 iterations, the final shape (at this poor precision) appears:

Indeed, much more work (150 iterations with ) is needed for disclosing with finer detail the solution:

The solution settles in a regime of slowly damped oscillations about the limit value of 0.1078 (indicated by a thin horizontal line). We have also shown in dashed line the analytical solution of the (continuous) problem at according to Boucekkine et al.(1997a). In the linear utility case, a constant optimal scrapping arises, equal say to . With the same “too high” initial function , , equilibrium on the labor market at requires a scrapping value such that . So the interior solution is not implementable from . The adjustment to the optimal solution takes the form of the corner solution since the economy begins with “too much” capital. Once the optimal scrapping value is reached, the economy settles in a regime of undamped fluctuations corresponding to everlasting replacement echoes, as shown in the figure above. When , the economy starts investing before and oscillations are damped: This is obviously due to consumption smoothing when utility functions are strictly concave. Much smoother solution paths can be obtained for lower 's values, but we have preferred a parameterization close to the linear utility case and initial conditions very far from the corresponding balanced growth paths so as to illustrate the ability of our method to reproduce (highly) irregular solution paths.

[^{7}T^{*}]

and

is indeed the fixed point of function . With and , optimal scrapping is around 9. With the initial condition, , t < 0, the constraint (3) of the optimization problem implies since is around 3.3.

5 Concluding remarks

In this paper, we have built up a Gauss-Seidel scheme in order to solve vintage capital models, especially those with Leontieff technologies. Using theoretical and experimental arguments, we have shown why our approach is suitable to deal with this class of models. In particular, the use of a cyclic coordinate descent optimization technique within our procedure seems adapted to circumvent the extremely arduous problems coming from the complex structure of the optimality conditions of these models. It should be noted that our method is quite general, and can be applied to a much larger set of dynamic models than those referenced in this paper. Furthermore, there is room for many potential improvements and adaptations of the method depending on the concrete problem under consideration. We have just presented a basic algorithmic scheme, which is flexible to a large extent. Clearly, the simple numerical integration technique we have adopted for convenience in this paper can be replaced by a more complex and accurate method. Also, the elementary relaxation scheme we have implemented so far can be replaced by more efficient first-order iterations like successive over relaxation (SOR) processes. Practitioners can take advantage of this flexibility to optimize their numerical setting if necessary. This is indeed a major strength of our method.

Indeed, except the updating of state variables using the constraints of the considered optimization problem, which is model specific, the rest of the method is absolutely general.

6 References

1. Aghion, Ph. and P. Howitt, 1994, Growth and unemployment, Review of Economic Studies 61, 477-494.
2. Baker, C. and C. Paul, 1996, A global convergence theorem for a class of parallel explicit Runge-Kutta methods and vanishing lag delay differential equations, SIAM Journal of Numerical Analysis 33, 1559-1576.
3. Benhabib, J and A. Rustichini, 1993, A vintage capital model of investment and growth, in R. Becker et al. Eds, General Equilibrium, Growth and Trade. II The Legacy of Lionel McKenzie (Academic Press, New York) 248-301.
4. Boucekkine, R., M. Germain and O. Licandro, 1997a, Replacement echoes in the vintage capital growth models, Journal of Economic Theory 74, 333-348.
5. Boucekkine, R., O. Licandro and C. Paul, 1997b, Differential-difference equations in economics: On the numerical solution of vintage capital growth models, Journal of Economic Dynamics and Control 21, 347-362.
6. Boucekkine, R., M. Germain, O. Licandro and A. Magnus, 1998, Creative destruction, investment volatility and the average age of capital, Journal of Economic Growth, forthcoming.
7. Caballero, R. and M. Hammour, 1996, On the timing and efficiency of creative destruction, Quarterly Journal of Economics 111, 805-851.
8. Caballero, R. and M. Hammour, 1998, Jobless growth: Appropriability, factor substitution, and unemployment, Carnegie-Rochester Conference Series on Public Policy 48, 51-94.
9. Luenberger, D., 1965, Introduction to linear and nonlinear programming (Addison-Wesley, Reading, Massachusetts).

COLECCION RESUMENES

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

TEXTOS EXPRESS

98-02: “Sector turístico y crecimiento del empleo en la Comunidad Autónoma de Canarias: Un ejercicio de prospección al horizonte 2011”, José A. Herce y Simón Sosvilla.

98-01: “El gasto sanitario en España: Evolución reciente y perspectiva”, Javier Alonso y José A. Herce.

DOCUMENTOS DE TRABAJO

98-20: “Numerical solution by iterative methods of a class of vintage capital models”, Raouf Boucekkine, Marc Germain, Omar Licandro y Alphonse Magnus.

98-19: “Endogenous vs exogeneously driven fluctuations in vintage capital models”, Raouf Boucekkine, Fernando del Río y Omar Licandro.

98-18: “Assesing the economic value of nearest-neighbour exchange-rate forecasts”, F. Fernández-Rodríguez, S. Sosvilla-Rivero y J. Andrada-Félix.

98-17: “Exchange-rate forecasts with simultaneous nearest-neighbour methods: Evidence from the EMS”, F. Fernández-Rodríguez, S. Sosvilla-Rivero y J. Andrada-Félix.

98-16: “Los efectos económicos de la Ley de Consolidación de la Seguridad Social. Perspectivas financieras del sistema tras su entrada en vigor”, José A. Herce y Javier Alonso.

98-15: “Economía, mercado de trabajo y sistema universitario español: Conversaciones con destacados macroeconomistas”, Carlos Usabiaga Ibáñez.

98-14: “Regional integration and growth: The Spanish case”, Ana Goicolea, José A. Herce y Juan J. de Lucio.

98-13: “Tax burden convergence in Europe”, Simón Sosvilla-Rivero, Miguel Angel Galindo y Javier Alonso.

98-12: “Growth and the Welfare State in the EU: A cusality analysis”, José A. Herce, Simón Sosvilla-Rivero y Juan J. de Lucio.

98-11: “Proyección de la población española 1991-2026. Revisión 1997”, Juan Antonio Fernández Cordón.

98-10: “A time-series examination of convergence in social protection across EU countries”, José A. Herce, Simón Sosvilla-Rivero y Juan J. de Lucio.

98-09: “Estructura Demográfica y Sistemas de Pensiones. Un análisis de equilibrio general aplicado a la economía española”, María Montero Muñoz.

98-08: “Earnings inequality in Portugal and Spain: Contrasts and similarities”, Olga Cantó, Ana R. Cardoso y Juan F. Jimeno.

98-07: “Labour reallocation and labour market institutions: Evidence from Spain”, Carlos García-Serrano y Juan F. Jimeno.

98-06: “Benchmark priors for Bayesian model averaging”, Carmen Fernández, Eduardo Ley, Mark F. J. Steel.

98-05: “The effects of externalities on value added and productivity growth in Spanish industry”, Juan J. de Lucio, José A. Herce y Ana Goicolea.

98-04: “Employment segmentation, labour mobility and Mismatch: Spain, 1987-1993”, Sonsoles Castillo, Juan F. Jimeno y Omar Licandro.

98-03: “Un análisis global, regional y sectorial de los efectos externos de conocimiento”, Juan J. de Lucio.

98-02: “A tale of two neighbour economies: Does wage flexibility make the difference between Portuguese and Spanish unemployment?”, Sonsoles Castillo, Juan J. Dolado y Juan F. Jimeno.

98-01: “Social protection benefits and growth: Evidence from the European Union”, José A. Herce, Simón Sosvilla y Juan J. de Lucio.