Directory UMM :Data Elmu:jurnal:J-a:Journal Of Economic Dynamics And Control:Vol25.Issue5.Apr2001:
Journal of Economic Dynamics & Control
25 (2001) 655}669
Numerical solution by iterative methods
of a class of vintage capital modelsq
Raouf Boucekkine!,*, Marc Germain", Omar Licandro#,
Alphonse Magnus$
!Universite& Catholique de Louvain, IRES, Place Montesquieu, 3, B-1348 Louvain-la-Neuve, Belgium
"CORE, Louvain-la-Neuve, Belgium
#FEDEA, Madrid, Spain
$UCL, Louvain-la-Neuve, Belgium
Received 15 December 1997; accepted 2 September 1999
Abstract
We build up an iterative numerical procedure in order to solve vintage capital
growth models with nonlinear utility functions and Leontie! technologies, a class
of models intensively used in the literature since the early 1990s. The numerical procedure
is of the relaxation type and uses a step-by-step maximization scheme for updating.
The procedure is close to the cyclic coordinate descent algorithm as described in the
computational mathematics literature. We explain why and how our numerical scheme is
suitable to handle the considered class of models. ( 2001 Elsevier Science B.V. All
rights reserved.
JEL classixcation: C63; E32; O40
Keywords: Vintage capital models; Relaxation; Optimization; Cyclic coordinate descent
algorithm
q
We are extremely grateful to Ken Judd and to Luis Puch for their most stimulating suggestions.
The numerical computations have been performed on the Convex Exemplar SPP-1600 of the
UniversiteH Catholique de Louvain. R. Boucekkine has been supported by a Grant &Actions de
Recherche ConcerteH c' of the Ministry of Scienti"c Research of the Belgian French speaking
community.
* Corresponding author. Tel.: #32 10 47 39 48; fax: #32 10 47 39 45.
E-mail address: boucekkine@ires.ucl.ac.be (R. Boucekkine).
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 5 6 - 1
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R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
1. Introduction
The analysis of vintage capital growth models has regained interest since the
early 1990s. Indeed, this kind of models allows 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.1 Investment volatility and equipment replacement are
analyzed using vintage capital models by Benhabib and Rustichini (1993) and by
Boucekkine et al. (1997a, 1998, 1999). Using the same framework, Aghion and
Howitt (1994) and Caballero and Hammour (1996), among others, have analyzed the e!ects 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 a Leontie! 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 in"nite
optimal lifetimes for equipment, which sounds unrealistic. Moreover, when the
utility functions are linear, Leontie! technologies allow to bring out some
analytical results as in Boucekkine et al. (1997a).
When the utility functions are nonlinear, numerical resolution is unavoidable.
The major di$culty in handling numerically optimal growth vintage capital
models with the latter feature and with Leontie! technologies, results in the
treatment of the endogenous leads appearing in the associated optimality
necessary conditions. Precisely, these conditions together with Leontie! speci"cations yield systems of di!erential-di!erence or integro-di!erential-di!erence
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; 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 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.
1 Creative destruction re#ects the idea that economic growth is mainly guided by successive
innovations leading to the &death' of the resulting less productive techniques and equipment.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
657
As the latter treatment seems at best only implementable on di!erentialdi!erence 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
"xed-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 Leontie!
technology (like those quoted above for example). For convenience, we prefer to
focus on this class of models since the speci"cations 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-Leontie! 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
Leontie! 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
Leontie! 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 di$culties 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 nondi!erentiable 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 "rst 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
e$ciency and reliability 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.
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R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
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 Leontie! type, and the scrapping of old machines
is endogenous. The central planner is supposed to maximize social welfare by
solving the following problem:
P
max ="
=
u[c(t)] expM!otN dt
0
(1)
subject to
P
y(t)"
t
i(z) dz
(2)
t~T(t)
P
t
i(z) expM!czN dz"1
t~T(t)
y(t)"c(t)#i(t)
(3)
(4)
04i(t)4y(t)
given i(t) for all t(0. c(t) is consumption, y(t) is production, i(t) is investment
and ¹(t) represents the age of the oldest operating machines, or scrapping time.
The utility function u(c) is assumed to be increasing and strictly concave.
Parameter o is strictly positive and represents the rate of time preference.
Machines from vintage t, ∀t, are supposed to produce one unit of output each,
and do require exp(!ct) units of labor. The parameter c is positive and
represents (Harrod neutral) technical progress. Total labor resources are
assumed to be constant and equal to one. (2) is the Leontie! 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:2
t`J(t)
(1!ec(z~t~T(z)))u@(c(z))e~o(z~t) dz
(5)
t
with J(t)"¹(t#J(t)). J(t) 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
P
u@(c(t))"
2 All the properties of the model stated in this section are taken from Boucekkine et al. (1998).
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
659
while (2) and (3) capture its backward-looking dimension. Di!erentiating all
these equations with respect to time yields an integro-di!erential-di!erence
system with an endogenous lag ¹(t) and an endogenous lead J(t). 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 brie#y 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 pro"le 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.
3. Maximization by iteration: Numerical setting and theoretical justi5cations
3.1. The numerical setting
We decide to perform the approximate maximization of =, with
c(t)"y(t)!i(t) by (4), by replacing the unknown functions i and y by piecewise
constant functions on the intervals (0, D), (D, 2D),2 . Let i , i ,2; y , y ,2 be
0 1
0 1
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 u(y!i) still are piecewise
constant functions, the values of the latter one being u(y !i ), u(y !i ),2 .
0
0
1
1
Integrals of piecewise constant functions are computed exactly by the midpoint
rule:
(N`1)D
N
F(t) dt"D + F((k#1/2)D).
0
k/0
P
Remark that we keep a "nite sum, even if we have to discretize an integral
from 0 to R, in which case a large value is chosen for N. If we have to perform
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R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
Fig. 1. Computing the scrapping time.
an integral on an interval whose endpoints are not integer multiples of D, we
have to be careful with the "rst and the last terms:
P
b
a
F(t) dt"(k D!a)F((k !1/2)D)#DF((k #1/2)D)#2
1
1
1
#DF((k !1/2)D)#(b!k D)F((k #1/2)D),
2
2
2
where k D is the smallest integer multiple of D which is larger or equal than a,
1
and where k D is the largest integer multiple of D which is smaller or equal than
2
b (see the discussion based on Fig. 1).
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 D is small, and the exponential functions are slowly
varying.3 Hereafter we set t "(k#1/2)D, for k"0, 1,2, N, and we denote by
k
i and y the values of investment and output at t . Application of these
k
k
k
principles leads to the discrete-time problem analog to (1)}(4):
N
max = "D + u(y !i )exp(!ot ),
$*4#
k
k
k
k/0
subject to
(6)
D
y "( jD!t #¹(t ))i #Di #2#Di
# i , k"0, 1,2, N,
k
k
k j~1
j
k~1 2 k
(7)
where jD is the smallest integer multiple of D larger or equal than t !¹(t ),
k
k
1"( jD!t #¹(t ))i e~ctj~1 #Di e~ctj #2
j
k
k j~1
D
#Di
e~ctk~1 # i e~ctk , k"0, 1,2, N,
k~1
2 k
04i 4y , k"0, 1,2, N.
k
k
given i for all p(0.
p
(8)
3 Obviously, more accurate numerical integration procedures can be implemented at this stage
(see Judd (1998, Chapter 7) for a simple presentation of these procedures). We have not exploited so
far this line of research.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
661
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 N#1 numbers i , i ,2, i . To this end, we need the corresponding
0 1
N
numbers y , y ,2, y . These values of y are computed from (7) which is the
0 1
N
exact value of the integral (2) at t , k"0, 1,2, N, as far as the function i is
k
assumed to be piecewise constant. As the lower bound t !¹(t ) is normally not
k
k
an integer multiple of D, we have to consider the integer j such that
( j!1)D(t !¹(t )4jD. Fig. 1 makes it clear.
k
k
We see that the formula (7) represents indeed the sum of rectangular areas
between t and t !¹(t ). Well, everything is all right provided ¹(t ) is known.
k
k
k
k
This one is now computed from (3), according to the same principles, assuming
i(z) exp(!cz) reasonably close to a piecewise constant function. The formula (8)
is actually an equation for the unknown ¹(t )"¹ . We get the new area,
k
k
advancing from right to left, summing the successive rectangular areas
Di exp(!ct )/2, Di
exp(!ct
),
up to the point where we get a number,
k
k
k~1
k~1 2
say a larger than one, by summing the last small rectangle Di
exp(!ct ).
j~1
j~1
This means that we went too far to the left (that is how j is found) by an amount
a!1. The left endpoint of the integral is therefore not ( j!1)D, but
t !¹(t )"( j!1)D#(a!1)/[i
exp(!ct )],
k
k
j~1
j~1
yielding the computed value of ¹(t ). The maximization (6) is performed by
k
iteration. At each iteration, we use a step-by-step maximization device. Precisely, at each step, we look at the in#uence of a single value i associated to a time
k
t "(k#1/2)D on the discretized integral = , keeping unchanged the other
k
$*4#
investment ordinates. To ease the presentation of the algorithm, we will write
Table 1
Numerical solution by iterative methods of a class of vintage capital models
The resolution algorithm
(a) Initialization
Choose an initial investment vector I0"[i (0), i(0),2, i(0)]@ and a stopping
0 1
N
parameter e'0. Compute the induced scrapping time and output vectors,
¹0 and >0, using (7) and (8).
(b) Maximization step by step (Gauss}Seidel step)
Step 0. Solve i(1)"argmax = (i , i(0), i(0),2, i(0)). Update ¹0 and >0
N
0
i0 $*4# 0 1 2
using (7) and (8).
Step k. For k"1, 2,2, N!1, solve i (1)"argmax = (i(1),2, i(1) , i , i(0) ,2, i(0)).
k
ik $*4# 0
k~1 k k`1
N
Update ¹0 and >0 using (7) and (8).
Step N. Solve i(1)"argmax = (i (1), i(1),2, i(1) , i ). Form I1"[i (1), i(1),2, i(1)]@
N~1 N
0 1
N
N
iN $*4# 0 1
and the induced updated scrapping time and output vectors ¹1 and >1.
(c) Gauss}Seidel iteration
Given a vector norm D . D, if DI1!I0D/DI0D(e, STOP and return I0; else set I0 equal
to I1 and go to steps (b) and (c).
662
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
= as a function of the sole investment vector ordinates, = (i , i ,2, i ),
$*4#
$*4# 0 1
N
since the output and scrapping time ordinates are computed from the investment vector thanks to Eqs. (7) and (8). The whole algorithm works as reported in
Table 1.
As one can see, we follow basically a Gauss}Seidel approach for the location
of the optimal solution paths. At each Gauss}Seidel iteration, a step-by-step
maximization is performed. At each step k of iterations (b), the discretized
integral is maximized with respect to the ordinate i of the investment vector,
k
, if any, with
keeping unchanged future investment ordinates i
, i
k`1 k`2 2
respect to the base, with the preceding investment ordinates i , 04l4k!1,
l
updated thanks to the already performed maximization steps. However, even
if we abstract away from the required optimization work, the implemented
maximization device and the need to recompute y and ¹ 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 behavior of the error through iterations (c).4
It should be noted, that except the required (and costly) updating of y and
¹ at each maximization step, our steps (b) and (c) correspond to the cyclic
coordinate descent optimization algorithm as described in Luenberger (1965,
pp. 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.5 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, [i i 2 i ]@, when N is large, the cyclic descent algorithm
0 1
N
appears very attractive for our purposes. Although this algorithm is known to
have slower convergence rates than certain gradient-based optimization
methods, we "nd 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.
3.2. Theoretical justixcations 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
4 Indeed, in our most re"ned tests, we took D"0.1, about 1% of the average value of the
scrapping time ¹, and a time solution horizon equal to 100, about 10 times the average ¹. So, we had
about 103 one-dimensional maximizations to perform per iteration. Finally, at least 100 iterations (c)
are needed to locate the "xed-point for a convergence tolerance level around 10~4.
5 As the popular steepest descent algorithm, see again Luenberger (1965, pp. 148}154).
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
663
a di!erential 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
$*4#
as a function of a single ordinate i(t ), for any xxed t 50.
0
0
Proof. Indeed, let =(a) be the value of = in a situation where i(t )"a, where
$*4#
0
t "(k#1/2)D, D a su$ciently small number and k is some nonnegative
0
integer.
We now decide to give some new value i(t )"b on the interval of width D and
0
center t , with i(t) unchanged in all the other intervals. The value of =(b)
0
depends on all the changes induced in the various values c(( j#1/2)D). Let
t "( j#1/2)D be another time abscissa. We have c(t )"y(t )!i(t ), so we
1
1
1
1
have only to appreciate the change in y(t ), which depends on ¹(t ). From (8),
1
1
t1
t1
i(z) e~cz dz"
i(z) e~cz dz#(b!a)De~ct0 ,
1"
b
a
t1 ~T(t1 )b
t1 ~T(t1 )b
only if t !¹(t )4t 4t , as (8) will be kept unchanged in other cases.6 Let us
1
1
0
1
subtract the &old' equation (8):
P
P
P
t1 ~T(t1 )a
i(z)e~cz dz#(b!a)De~ct0
t1 ~T(t1 )b
+[¹(t ) !¹(t ) ]i(t !¹(t ))e~c(t1 ~T(t1 )#(b!a)De~ct0 ,
1b
1a 1
1
if b is chosen to be close to a, so that the change of ¹(t ) is small.
1
We now look at the new value of y(t ):
1
t1
t1
y(t ) "
i(z) dz"
i(z) dz#(b!a)D,
1b
b
a
1
1
b
1
1
b
t ~T(t )
t ~T(t )
t1 ~T(t1 )a
i(z) dz#(b!a)D
y(t ) !y(t ) "
a
1b
1a
t1 ~T(t1 )b
+[¹(t ) !¹(t ) ]i(t !¹(t ))#(b!a)D,
1b
1a 1
1
which simpli"es into
0"
P
(9)
P
P
y(t ) !y(t ) "(b!a)D[1!exp(c(t !¹(t )!t ))],
1b
1a
1
1
0
6 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.
664
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
thanks to (9). We just got the increase of c(t )"y(t )!i(t ), as far as
1
1
1
t !¹(t )4t (t . At t , the change is of course
1
1
0
1
0
c(t ) !c(t ) "y(t )!b![y(t )!a]"a!b,
0b
0a
0
0
and the change for = is
=(b)!=(a)
"D[u(c(t ) #a!b)!u(c(t ) ]e~ot0
0a
0a
#D
+
Mu(c(t )
1a
t1 ~T(t1 )yt0 :t1
#(b!a)D[1!ec(t1 ~T(t1 )~t0 )])! u(c(t ) )Ne~ot1 ,
1a
(10)
for b close to a and for small D.
As (10) is a linear combination of concave functions of b, =(b) is indeed
a concave function too, for b in a small interval, but concavity must only be
established locally to hold everywhere. h
As = is now known to be a concave function of a single i ordinate, we are
$*4#
therefore sure that = (x) has a unique maximum in the admissible range, i.e.,
$*4#
from x"0 up to the value such that x"i(t )"y(t ). Provided that all the
0
0
di!erentiations are valid (or in other words that an interior solution exists for
the optimization problem at t ), an interesting consequence of (10) is the
0
di!erential relation when bPa"x:
C
=@ (x)"D !u@(c(t ))e~ot0
$*4#
0
P
#
t0 `J(t0 )
t0
D
u@(c(t ))[1!ec(t1 ~T(t1 )~t0 )]e~ot1 dt ,
1
1
(11)
and =@ (x)"0 for all t 50 yields the well known optimality condition (5),
$*4#
0
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 "rst-order conditions when an interior
solution is encountered.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
665
4. An example
Let us solve the problem with o"0.05, c"0.03, u(c)"ch, h"0.85 close to 1.
So we have a weakly concave function to maximize, and we expect rather
a rough behavior of the solution paths. The balanced growth path solution is
i(t)"0.1078 exp(0.03t): if this law holds for t(0, it will still hold when t50. As
one expects a perturbation of this law, one makes the computations with
i approximating i(t) exp(!ct) at t"(k#1/2)D, k"0, 1,2
k
One decides to take i(t)"0.3 exp(0.03t) 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 i(t)"0.1078 exp(0.03t) at t'0, i.e.,
i "0.1078 for all k50. Let us perform the approximate calculations with
k
a (much too large) time step D"1. The results of the step by step maximization
for i , i , i and i are reported in Table 2. First, we look at the discretized
0 1 3
4
integral =
for several test values i , keeping i , i , etc. unchanged. The
$*4#
0
1 2
discretized integral is the largest when i "0, which is not surprising, as i(t) was
0
too large at t(0. We keep i "0, and make i move. Again, i "0 is best. And
0
1
1
we get the same answer for i . Setting i "i "i "0, we maximize with
2
0
1
2
respect to i . Some change at last, the best i appears to be close to 0.11. We now
3
3
look at i . Here, the best value of i is about 0.18.
4
4
One goes on with next values (in order not to be bothered by truncated series
e!ects in the estimation of the integral, we took a big upper bound, N"500).
These successive maximizations of =
(steps (b) using the terminology of
$*4#
Section 3.1) lead to the "rst approximation to i(t) exp(!ct). The job is not
"nished yet! We just performed maximization of =
on each i , keeping
$*4#
k
unchanged the i 's for j(k. This is only a limited way to see how i interacts
j
k
with its neighbors, 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 "rst 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 simpli"ed calculations was achieved for all parts,
there was still no exact matching, and the whole process was restarted again,
and again.
On our example, a second iteration (or a second run of maximizations) does
not change the values of i , i , and i , which are still kept at the zero level.
0 1
2
However, the best i is found to be no more 0.11, but 0.04. Things hardly
3
changed. After about 20 iterations, the "nal shape (at this poor D"1 precision)
appears. Fig. 2 displays the patterns of (detrended) investment obtained after the
"rst, the second, and the twentieth iteration of type (c).
After the "rst Gauss}Seidel iteration, most of the values stay close to the
initial 0.1078. One sees slight depressions at t"11 and t"19, faint echoes
of the big depression near the origin. But since the time discretization is too
666
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
Table 2
Numerical solution by iterative methods of a class of vintage capital models, results of the "rst
maximization step with respect to i , i , i and i
0 1 3
4
P
P
P
P
i
0
0
0.05
0.1
0.15
0.2
=
$*4#
32.7516401
32.7313349
32.7089774
32.6845603
32.6588714
i
1
0
0.05
0.1
0.15
0.2
=
$*4#
32.778625
32.7673855
32.7539278
32.7379688
32.7206465
i
3
0
0.05
0.1
0.11
0.12
0.15
=
$*4#
32.7801023
32.7851276
32.787636
32.7876691
32.7875742
32.786856
i
4
0
0.05
0.1
0.15
0.16
0.17
0.18
0.19
=
$*4#
32.7758335
32.7825734
32.7873551
32.7884092
32.7884577
32.7884902
32.7885064
32.7884807
big (D"1), the obtained paths are not su$ciently &"ne', even after 20 iterations.
Indeed, much more work (150 iterations with D"0.2) is needed for disclosing
with "ner detail the solution. The obtained solution is reported in Fig. 3. The
solution settles in a regime of slowly damped oscillations about the limit value of
0.1078 (indicated by a thin horizontal line in Fig. 3). We have also shown in
dashed line the analytical solution of the (continuous) problem at h"1 according to Boucekkine et al. (1997a).
In the linear utility case, a constant optimal scrapping arises, equal say to
¹H. With the same &too high' initial function i(t)"0.3 exp(0.03t), t(0, equilibrium on the labor market at t"0 requires a scrapping value ¹(0) such that
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
667
Fig. 2. (a) The investment pattern after one Gauss}Seidel iteration for D"1. (b) The investment
pattern after 2 Gauss}Seidel iterations for D"1. (c) The investment pattern after 20 Gauss}Seidel
iterations for D"1.
Fig. 3. The investment pattern after 150 Gauss}Seidel iterations for D"0.2.
¹(0)(¹H.7 So the interior solution is not implementable from t"0. The
adjustment to the optimal solution takes the form of the corner solution i(t)"0
since the economy begins with &too much' capital. Once the optimal scrapping
value ¹H is reached, the economy settles in a regime of undamped #uctuations
7 ¹H is indeed the "xed point of function F(x)"!(1/c) ¸n(1!o#c!c/o#(c/o) exp(!ox)).
With c"0.03 and o"0.05, optimal scrapping ¹H is around 9. With the initial condition,
i(t)"0.3 exp(0.03t), t(0, the constraint (3) of the optimization problem implies ¹(0)(¹H since
¹(0) is around 3.3.
668
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
corresponding to everlasting replacement echoes, as shown in the "gure above.
When h"0.85, 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 h'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.
5. Concluding remarks
In this paper, we have built up a Gauss}Seidel scheme in order to solve
vintage capital models, especially those with Leontie! 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.8 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 #exible 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 e$cient "rst-order iterations like successive over relaxation
(SOR) processes. Practitioners can take advantage of this #exibility to optimize
their numerical setting if necessary. This is indeed a major strength of
our method.
References
Aghion, Ph., Howitt, P., 1994. Growth and unemployment. Review of Economic Studies 61,
477}494.
Baker, C., Paul, C., 1996. A global convergence theorem for a class of parallel explicit Runge}Kutta
methods and vanishing lag delay di!erential equations. SIAM Journal of Numerical Analysis 33,
1559}1576.
8 Indeed, except the updating of state variables using the constraints of the considered optimization problem, which is model speci"c, the rest of the method is absolutely general.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
669
Benhabib, J., Rustichini, A., 1993. A vintage capital model of investment and growth. In: Becker, R.
et al. (Eds.), General Equilibrium, Growth and Trade. II The Legacy of Lionel McKenzie.
Academic Press, New York, 248}301.
Boucekkine, R., Germain, M., Licandro, O., 1997a. Replacement echoes in the vintage capital
growth models. Journal of Economic Theory 74, 333}348.
Boucekkine, R., Licandro, O., Paul, C., 1997b. Di!erential-di!erence equations in economics: on the
numerical solution of vintage capital growth models. Journal of Economic Dynamics and
Control 21, 347}362.
Boucekkine, R., Germain, M., Licandro, O., Magnus, A., 1998. Creative destruction, investment
volatility and the average age of capital. Journal of Economic Growth 3, 361}384.
Boucekkine, R., del RmH o, F., Licandro, O., 1999. Endogenous Vs exogenously driven #uctuations in
vintage capital models. Journal of Economic Theory 88, 161}187.
Caballero, R., Hammour, M., 1996. On the timing and e$ciency of creative destruction. Quarterly
Journal of Economics 111, 805}851.
Caballero, R., Hammour, M., 1998. Jobless growth: appropriability, factor substitution, and unemployment. Carnegie}Rochester Conference Series on Public Policy 48, 51}94.
Judd, K., 1998. Numerical Methods in Economics. The MIT Press, Cambridge, MA.
Luenberger, D., 1965. Introduction to Linear and Nonlinear Programming. Addison-Wesley,
Reading, MA.
25 (2001) 655}669
Numerical solution by iterative methods
of a class of vintage capital modelsq
Raouf Boucekkine!,*, Marc Germain", Omar Licandro#,
Alphonse Magnus$
!Universite& Catholique de Louvain, IRES, Place Montesquieu, 3, B-1348 Louvain-la-Neuve, Belgium
"CORE, Louvain-la-Neuve, Belgium
#FEDEA, Madrid, Spain
$UCL, Louvain-la-Neuve, Belgium
Received 15 December 1997; accepted 2 September 1999
Abstract
We build up an iterative numerical procedure in order to solve vintage capital
growth models with nonlinear utility functions and Leontie! technologies, a class
of models intensively used in the literature since the early 1990s. The numerical procedure
is of the relaxation type and uses a step-by-step maximization scheme for updating.
The procedure is close to the cyclic coordinate descent algorithm as described in the
computational mathematics literature. We explain why and how our numerical scheme is
suitable to handle the considered class of models. ( 2001 Elsevier Science B.V. All
rights reserved.
JEL classixcation: C63; E32; O40
Keywords: Vintage capital models; Relaxation; Optimization; Cyclic coordinate descent
algorithm
q
We are extremely grateful to Ken Judd and to Luis Puch for their most stimulating suggestions.
The numerical computations have been performed on the Convex Exemplar SPP-1600 of the
UniversiteH Catholique de Louvain. R. Boucekkine has been supported by a Grant &Actions de
Recherche ConcerteH c' of the Ministry of Scienti"c Research of the Belgian French speaking
community.
* Corresponding author. Tel.: #32 10 47 39 48; fax: #32 10 47 39 45.
E-mail address: boucekkine@ires.ucl.ac.be (R. Boucekkine).
0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 5 6 - 1
656
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
1. Introduction
The analysis of vintage capital growth models has regained interest since the
early 1990s. Indeed, this kind of models allows 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.1 Investment volatility and equipment replacement are
analyzed using vintage capital models by Benhabib and Rustichini (1993) and by
Boucekkine et al. (1997a, 1998, 1999). Using the same framework, Aghion and
Howitt (1994) and Caballero and Hammour (1996), among others, have analyzed the e!ects 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 a Leontie! 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 in"nite
optimal lifetimes for equipment, which sounds unrealistic. Moreover, when the
utility functions are linear, Leontie! technologies allow to bring out some
analytical results as in Boucekkine et al. (1997a).
When the utility functions are nonlinear, numerical resolution is unavoidable.
The major di$culty in handling numerically optimal growth vintage capital
models with the latter feature and with Leontie! technologies, results in the
treatment of the endogenous leads appearing in the associated optimality
necessary conditions. Precisely, these conditions together with Leontie! speci"cations yield systems of di!erential-di!erence or integro-di!erential-di!erence
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; 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 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.
1 Creative destruction re#ects the idea that economic growth is mainly guided by successive
innovations leading to the &death' of the resulting less productive techniques and equipment.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
657
As the latter treatment seems at best only implementable on di!erentialdi!erence 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
"xed-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 Leontie!
technology (like those quoted above for example). For convenience, we prefer to
focus on this class of models since the speci"cations 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-Leontie! 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
Leontie! 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
Leontie! 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 di$culties 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 nondi!erentiable 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 "rst 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
e$ciency and reliability 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.
658
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
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 Leontie! type, and the scrapping of old machines
is endogenous. The central planner is supposed to maximize social welfare by
solving the following problem:
P
max ="
=
u[c(t)] expM!otN dt
0
(1)
subject to
P
y(t)"
t
i(z) dz
(2)
t~T(t)
P
t
i(z) expM!czN dz"1
t~T(t)
y(t)"c(t)#i(t)
(3)
(4)
04i(t)4y(t)
given i(t) for all t(0. c(t) is consumption, y(t) is production, i(t) is investment
and ¹(t) represents the age of the oldest operating machines, or scrapping time.
The utility function u(c) is assumed to be increasing and strictly concave.
Parameter o is strictly positive and represents the rate of time preference.
Machines from vintage t, ∀t, are supposed to produce one unit of output each,
and do require exp(!ct) units of labor. The parameter c is positive and
represents (Harrod neutral) technical progress. Total labor resources are
assumed to be constant and equal to one. (2) is the Leontie! 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:2
t`J(t)
(1!ec(z~t~T(z)))u@(c(z))e~o(z~t) dz
(5)
t
with J(t)"¹(t#J(t)). J(t) 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
P
u@(c(t))"
2 All the properties of the model stated in this section are taken from Boucekkine et al. (1998).
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
659
while (2) and (3) capture its backward-looking dimension. Di!erentiating all
these equations with respect to time yields an integro-di!erential-di!erence
system with an endogenous lag ¹(t) and an endogenous lead J(t). 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 brie#y 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 pro"le 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.
3. Maximization by iteration: Numerical setting and theoretical justi5cations
3.1. The numerical setting
We decide to perform the approximate maximization of =, with
c(t)"y(t)!i(t) by (4), by replacing the unknown functions i and y by piecewise
constant functions on the intervals (0, D), (D, 2D),2 . Let i , i ,2; y , y ,2 be
0 1
0 1
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 u(y!i) still are piecewise
constant functions, the values of the latter one being u(y !i ), u(y !i ),2 .
0
0
1
1
Integrals of piecewise constant functions are computed exactly by the midpoint
rule:
(N`1)D
N
F(t) dt"D + F((k#1/2)D).
0
k/0
P
Remark that we keep a "nite sum, even if we have to discretize an integral
from 0 to R, in which case a large value is chosen for N. If we have to perform
660
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
Fig. 1. Computing the scrapping time.
an integral on an interval whose endpoints are not integer multiples of D, we
have to be careful with the "rst and the last terms:
P
b
a
F(t) dt"(k D!a)F((k !1/2)D)#DF((k #1/2)D)#2
1
1
1
#DF((k !1/2)D)#(b!k D)F((k #1/2)D),
2
2
2
where k D is the smallest integer multiple of D which is larger or equal than a,
1
and where k D is the largest integer multiple of D which is smaller or equal than
2
b (see the discussion based on Fig. 1).
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 D is small, and the exponential functions are slowly
varying.3 Hereafter we set t "(k#1/2)D, for k"0, 1,2, N, and we denote by
k
i and y the values of investment and output at t . Application of these
k
k
k
principles leads to the discrete-time problem analog to (1)}(4):
N
max = "D + u(y !i )exp(!ot ),
$*4#
k
k
k
k/0
subject to
(6)
D
y "( jD!t #¹(t ))i #Di #2#Di
# i , k"0, 1,2, N,
k
k
k j~1
j
k~1 2 k
(7)
where jD is the smallest integer multiple of D larger or equal than t !¹(t ),
k
k
1"( jD!t #¹(t ))i e~ctj~1 #Di e~ctj #2
j
k
k j~1
D
#Di
e~ctk~1 # i e~ctk , k"0, 1,2, N,
k~1
2 k
04i 4y , k"0, 1,2, N.
k
k
given i for all p(0.
p
(8)
3 Obviously, more accurate numerical integration procedures can be implemented at this stage
(see Judd (1998, Chapter 7) for a simple presentation of these procedures). We have not exploited so
far this line of research.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
661
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 N#1 numbers i , i ,2, i . To this end, we need the corresponding
0 1
N
numbers y , y ,2, y . These values of y are computed from (7) which is the
0 1
N
exact value of the integral (2) at t , k"0, 1,2, N, as far as the function i is
k
assumed to be piecewise constant. As the lower bound t !¹(t ) is normally not
k
k
an integer multiple of D, we have to consider the integer j such that
( j!1)D(t !¹(t )4jD. Fig. 1 makes it clear.
k
k
We see that the formula (7) represents indeed the sum of rectangular areas
between t and t !¹(t ). Well, everything is all right provided ¹(t ) is known.
k
k
k
k
This one is now computed from (3), according to the same principles, assuming
i(z) exp(!cz) reasonably close to a piecewise constant function. The formula (8)
is actually an equation for the unknown ¹(t )"¹ . We get the new area,
k
k
advancing from right to left, summing the successive rectangular areas
Di exp(!ct )/2, Di
exp(!ct
),
up to the point where we get a number,
k
k
k~1
k~1 2
say a larger than one, by summing the last small rectangle Di
exp(!ct ).
j~1
j~1
This means that we went too far to the left (that is how j is found) by an amount
a!1. The left endpoint of the integral is therefore not ( j!1)D, but
t !¹(t )"( j!1)D#(a!1)/[i
exp(!ct )],
k
k
j~1
j~1
yielding the computed value of ¹(t ). The maximization (6) is performed by
k
iteration. At each iteration, we use a step-by-step maximization device. Precisely, at each step, we look at the in#uence of a single value i associated to a time
k
t "(k#1/2)D on the discretized integral = , keeping unchanged the other
k
$*4#
investment ordinates. To ease the presentation of the algorithm, we will write
Table 1
Numerical solution by iterative methods of a class of vintage capital models
The resolution algorithm
(a) Initialization
Choose an initial investment vector I0"[i (0), i(0),2, i(0)]@ and a stopping
0 1
N
parameter e'0. Compute the induced scrapping time and output vectors,
¹0 and >0, using (7) and (8).
(b) Maximization step by step (Gauss}Seidel step)
Step 0. Solve i(1)"argmax = (i , i(0), i(0),2, i(0)). Update ¹0 and >0
N
0
i0 $*4# 0 1 2
using (7) and (8).
Step k. For k"1, 2,2, N!1, solve i (1)"argmax = (i(1),2, i(1) , i , i(0) ,2, i(0)).
k
ik $*4# 0
k~1 k k`1
N
Update ¹0 and >0 using (7) and (8).
Step N. Solve i(1)"argmax = (i (1), i(1),2, i(1) , i ). Form I1"[i (1), i(1),2, i(1)]@
N~1 N
0 1
N
N
iN $*4# 0 1
and the induced updated scrapping time and output vectors ¹1 and >1.
(c) Gauss}Seidel iteration
Given a vector norm D . D, if DI1!I0D/DI0D(e, STOP and return I0; else set I0 equal
to I1 and go to steps (b) and (c).
662
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
= as a function of the sole investment vector ordinates, = (i , i ,2, i ),
$*4#
$*4# 0 1
N
since the output and scrapping time ordinates are computed from the investment vector thanks to Eqs. (7) and (8). The whole algorithm works as reported in
Table 1.
As one can see, we follow basically a Gauss}Seidel approach for the location
of the optimal solution paths. At each Gauss}Seidel iteration, a step-by-step
maximization is performed. At each step k of iterations (b), the discretized
integral is maximized with respect to the ordinate i of the investment vector,
k
, if any, with
keeping unchanged future investment ordinates i
, i
k`1 k`2 2
respect to the base, with the preceding investment ordinates i , 04l4k!1,
l
updated thanks to the already performed maximization steps. However, even
if we abstract away from the required optimization work, the implemented
maximization device and the need to recompute y and ¹ 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 behavior of the error through iterations (c).4
It should be noted, that except the required (and costly) updating of y and
¹ at each maximization step, our steps (b) and (c) correspond to the cyclic
coordinate descent optimization algorithm as described in Luenberger (1965,
pp. 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.5 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, [i i 2 i ]@, when N is large, the cyclic descent algorithm
0 1
N
appears very attractive for our purposes. Although this algorithm is known to
have slower convergence rates than certain gradient-based optimization
methods, we "nd 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.
3.2. Theoretical justixcations 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
4 Indeed, in our most re"ned tests, we took D"0.1, about 1% of the average value of the
scrapping time ¹, and a time solution horizon equal to 100, about 10 times the average ¹. So, we had
about 103 one-dimensional maximizations to perform per iteration. Finally, at least 100 iterations (c)
are needed to locate the "xed-point for a convergence tolerance level around 10~4.
5 As the popular steepest descent algorithm, see again Luenberger (1965, pp. 148}154).
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
663
a di!erential 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
$*4#
as a function of a single ordinate i(t ), for any xxed t 50.
0
0
Proof. Indeed, let =(a) be the value of = in a situation where i(t )"a, where
$*4#
0
t "(k#1/2)D, D a su$ciently small number and k is some nonnegative
0
integer.
We now decide to give some new value i(t )"b on the interval of width D and
0
center t , with i(t) unchanged in all the other intervals. The value of =(b)
0
depends on all the changes induced in the various values c(( j#1/2)D). Let
t "( j#1/2)D be another time abscissa. We have c(t )"y(t )!i(t ), so we
1
1
1
1
have only to appreciate the change in y(t ), which depends on ¹(t ). From (8),
1
1
t1
t1
i(z) e~cz dz"
i(z) e~cz dz#(b!a)De~ct0 ,
1"
b
a
t1 ~T(t1 )b
t1 ~T(t1 )b
only if t !¹(t )4t 4t , as (8) will be kept unchanged in other cases.6 Let us
1
1
0
1
subtract the &old' equation (8):
P
P
P
t1 ~T(t1 )a
i(z)e~cz dz#(b!a)De~ct0
t1 ~T(t1 )b
+[¹(t ) !¹(t ) ]i(t !¹(t ))e~c(t1 ~T(t1 )#(b!a)De~ct0 ,
1b
1a 1
1
if b is chosen to be close to a, so that the change of ¹(t ) is small.
1
We now look at the new value of y(t ):
1
t1
t1
y(t ) "
i(z) dz"
i(z) dz#(b!a)D,
1b
b
a
1
1
b
1
1
b
t ~T(t )
t ~T(t )
t1 ~T(t1 )a
i(z) dz#(b!a)D
y(t ) !y(t ) "
a
1b
1a
t1 ~T(t1 )b
+[¹(t ) !¹(t ) ]i(t !¹(t ))#(b!a)D,
1b
1a 1
1
which simpli"es into
0"
P
(9)
P
P
y(t ) !y(t ) "(b!a)D[1!exp(c(t !¹(t )!t ))],
1b
1a
1
1
0
6 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.
664
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
thanks to (9). We just got the increase of c(t )"y(t )!i(t ), as far as
1
1
1
t !¹(t )4t (t . At t , the change is of course
1
1
0
1
0
c(t ) !c(t ) "y(t )!b![y(t )!a]"a!b,
0b
0a
0
0
and the change for = is
=(b)!=(a)
"D[u(c(t ) #a!b)!u(c(t ) ]e~ot0
0a
0a
#D
+
Mu(c(t )
1a
t1 ~T(t1 )yt0 :t1
#(b!a)D[1!ec(t1 ~T(t1 )~t0 )])! u(c(t ) )Ne~ot1 ,
1a
(10)
for b close to a and for small D.
As (10) is a linear combination of concave functions of b, =(b) is indeed
a concave function too, for b in a small interval, but concavity must only be
established locally to hold everywhere. h
As = is now known to be a concave function of a single i ordinate, we are
$*4#
therefore sure that = (x) has a unique maximum in the admissible range, i.e.,
$*4#
from x"0 up to the value such that x"i(t )"y(t ). Provided that all the
0
0
di!erentiations are valid (or in other words that an interior solution exists for
the optimization problem at t ), an interesting consequence of (10) is the
0
di!erential relation when bPa"x:
C
=@ (x)"D !u@(c(t ))e~ot0
$*4#
0
P
#
t0 `J(t0 )
t0
D
u@(c(t ))[1!ec(t1 ~T(t1 )~t0 )]e~ot1 dt ,
1
1
(11)
and =@ (x)"0 for all t 50 yields the well known optimality condition (5),
$*4#
0
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 "rst-order conditions when an interior
solution is encountered.
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
665
4. An example
Let us solve the problem with o"0.05, c"0.03, u(c)"ch, h"0.85 close to 1.
So we have a weakly concave function to maximize, and we expect rather
a rough behavior of the solution paths. The balanced growth path solution is
i(t)"0.1078 exp(0.03t): if this law holds for t(0, it will still hold when t50. As
one expects a perturbation of this law, one makes the computations with
i approximating i(t) exp(!ct) at t"(k#1/2)D, k"0, 1,2
k
One decides to take i(t)"0.3 exp(0.03t) 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 i(t)"0.1078 exp(0.03t) at t'0, i.e.,
i "0.1078 for all k50. Let us perform the approximate calculations with
k
a (much too large) time step D"1. The results of the step by step maximization
for i , i , i and i are reported in Table 2. First, we look at the discretized
0 1 3
4
integral =
for several test values i , keeping i , i , etc. unchanged. The
$*4#
0
1 2
discretized integral is the largest when i "0, which is not surprising, as i(t) was
0
too large at t(0. We keep i "0, and make i move. Again, i "0 is best. And
0
1
1
we get the same answer for i . Setting i "i "i "0, we maximize with
2
0
1
2
respect to i . Some change at last, the best i appears to be close to 0.11. We now
3
3
look at i . Here, the best value of i is about 0.18.
4
4
One goes on with next values (in order not to be bothered by truncated series
e!ects in the estimation of the integral, we took a big upper bound, N"500).
These successive maximizations of =
(steps (b) using the terminology of
$*4#
Section 3.1) lead to the "rst approximation to i(t) exp(!ct). The job is not
"nished yet! We just performed maximization of =
on each i , keeping
$*4#
k
unchanged the i 's for j(k. This is only a limited way to see how i interacts
j
k
with its neighbors, 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 "rst 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 simpli"ed calculations was achieved for all parts,
there was still no exact matching, and the whole process was restarted again,
and again.
On our example, a second iteration (or a second run of maximizations) does
not change the values of i , i , and i , which are still kept at the zero level.
0 1
2
However, the best i is found to be no more 0.11, but 0.04. Things hardly
3
changed. After about 20 iterations, the "nal shape (at this poor D"1 precision)
appears. Fig. 2 displays the patterns of (detrended) investment obtained after the
"rst, the second, and the twentieth iteration of type (c).
After the "rst Gauss}Seidel iteration, most of the values stay close to the
initial 0.1078. One sees slight depressions at t"11 and t"19, faint echoes
of the big depression near the origin. But since the time discretization is too
666
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
Table 2
Numerical solution by iterative methods of a class of vintage capital models, results of the "rst
maximization step with respect to i , i , i and i
0 1 3
4
P
P
P
P
i
0
0
0.05
0.1
0.15
0.2
=
$*4#
32.7516401
32.7313349
32.7089774
32.6845603
32.6588714
i
1
0
0.05
0.1
0.15
0.2
=
$*4#
32.778625
32.7673855
32.7539278
32.7379688
32.7206465
i
3
0
0.05
0.1
0.11
0.12
0.15
=
$*4#
32.7801023
32.7851276
32.787636
32.7876691
32.7875742
32.786856
i
4
0
0.05
0.1
0.15
0.16
0.17
0.18
0.19
=
$*4#
32.7758335
32.7825734
32.7873551
32.7884092
32.7884577
32.7884902
32.7885064
32.7884807
big (D"1), the obtained paths are not su$ciently &"ne', even after 20 iterations.
Indeed, much more work (150 iterations with D"0.2) is needed for disclosing
with "ner detail the solution. The obtained solution is reported in Fig. 3. The
solution settles in a regime of slowly damped oscillations about the limit value of
0.1078 (indicated by a thin horizontal line in Fig. 3). We have also shown in
dashed line the analytical solution of the (continuous) problem at h"1 according to Boucekkine et al. (1997a).
In the linear utility case, a constant optimal scrapping arises, equal say to
¹H. With the same &too high' initial function i(t)"0.3 exp(0.03t), t(0, equilibrium on the labor market at t"0 requires a scrapping value ¹(0) such that
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
667
Fig. 2. (a) The investment pattern after one Gauss}Seidel iteration for D"1. (b) The investment
pattern after 2 Gauss}Seidel iterations for D"1. (c) The investment pattern after 20 Gauss}Seidel
iterations for D"1.
Fig. 3. The investment pattern after 150 Gauss}Seidel iterations for D"0.2.
¹(0)(¹H.7 So the interior solution is not implementable from t"0. The
adjustment to the optimal solution takes the form of the corner solution i(t)"0
since the economy begins with &too much' capital. Once the optimal scrapping
value ¹H is reached, the economy settles in a regime of undamped #uctuations
7 ¹H is indeed the "xed point of function F(x)"!(1/c) ¸n(1!o#c!c/o#(c/o) exp(!ox)).
With c"0.03 and o"0.05, optimal scrapping ¹H is around 9. With the initial condition,
i(t)"0.3 exp(0.03t), t(0, the constraint (3) of the optimization problem implies ¹(0)(¹H since
¹(0) is around 3.3.
668
R. Boucekkine et al. / Journal of Economic Dynamics & Control 25 (2001) 655}669
corresponding to everlasting replacement echoes, as shown in the "gure above.
When h"0.85, 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 h'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.
5. Concluding remarks
In this paper, we have built up a Gauss}Seidel scheme in order to solve
vintage capital models, especially those with Leontie! 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.8 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 #exible 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 e$cient "rst-order iterations like successive over relaxation
(SOR) processes. Practitioners can take advantage of this #exibility to optimize
their numerical setting if necessary. This is indeed a major strength of
our method.
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