Echoes Dynamics in Vintage Models: Basic Theoretical and Computational Results

Raouf Boucekkine and Omar Licandro - Universidad Carlos III de Madrid

Echoes effects refer to the ability of an economy to reproduce its own past behaviour, mainly through the replacement activities of the old capital goods by the capital goods incorporating the latest and most efficient technology. Since the beginning of this decade, much attention has been paid to echoes effects as an important source of economic fluctuations. An early contribution of this field is due to Benhabib and Rustichini (1991) who stressed the importance of the replacement activities in the explanation of investment volatility. From the theoretical point of view, echoes effects can be considered as a third cause of economic fluctuations, to be added to the two traditional fluctuations sources: nonlinearities and non-stationary environments. However, unlike the latter sources, echoes cannot arise in models with homogenous goods and require a vintage formulation.

The vintage specification is currently very popular among economists, especially because it allows to reproduce some stylized facts (recently outlined by Gordon (1990)) that cannot be recovered within the standard one-sector neoclassical growth models. However, the mathematical and computational treatment of vintage models is by far more complicated than the treatment of the models with homogenous goods. This makes highly problematic the analysis of echoes effects. When the vintage models additionally include nonlinearities and/or non-stationary environments (see for example, Caballero and Hammour (1996)), it is not clear at all which part of the cyclical dynamics is due to echoes and which part is the result of the traditional fluctuations sources. This feature clearly complicates the economic interpretation of the obtained dynamics, although many authors seem to omit this issue.

In fact, it seems quite impossible to evaluate accurately which part of the dynamics is due to any of the three fluctuations sources, simply because that these sources strongly interact, the resulting dynamics being a non-trivial combination of the effects generated by each source. In this paper, we adopt a simpler approach. We focus on the non-traditional fluctuations source, namely the replacement echoes. To make clear the importance of echoes fluctuations, we first analyze continuous time canonical growth models either of the Solow type (ie. with constant saving rates) or of the Ramsey type with linear utility functions, incorporating Leontieff production functions and stationary environments. This will allow us to find out some useful properties as to the occurrence and strenght of echoes effects in this class of "linear" deterministic models. A single deviation with respect to this class of models is studied through the analysis of the case of strictly concave utility functions. As stressed by Benhabib and Rustichini (1991), introducing strict concavity in these models through the utility function will not have strong effects on the benchmark dynamics in the case of near-linear utility models. We numerically investigate this property and we bring out some "qualitative" conclusions on the interaction of nonlinearities and echoes effects in vintage capital growth models with Leontieff technologies.

The paper is organized as follows:

Section 1: The Solow vintage capital growth models
In this section, we numerically analyze the vintage capital growth model described in Solow et alii (1966). As in any Solow growth model, the saving rate is constant. The technology is Leontieff and the (Harrod-neutral) deterministic technical progress grows at a constant rate. The model yields a differential-difference equations system with an endogenous delay. The latter system is solved using an advanced method of steps a la Bellman and Cooke (1963). The method and the model example are also considered in Boucekkine, Licandro and Paul (1997). We re-interpret the results of this paper in terms of echoes effects (omitted in the latter paper) and complete the re-oriented analysis by performing further simulations. We conclude that in the Solow model, echoes effects disappear in the long run (as theoretically established by Solow et alii (1966)) and are extremely weak in the short run, whatever is the investment initial profile.

Section 2: The linear Ramsey vintage capital growth model
This section closely builds on Boucekkine, Germain and Licandro (1996). In the class of optimal growth models, the case with linear utility allows for an analytical resolution. We show that after a finite time adjustment period, depending on the investment initial profile, echoes effects appear and are ever-lasting. Starting at a finite date, detrended investment is forever periodic. The linear Ramsey model can be analytically handled because it gives rise to a recursive purely forward-looking block that can be solved using a non-standard fixed-point argument. Indeed, the latter recursive block can be transformed into a bivariate differential-difference system with an endogenous lead, by contrast to the Solow models mentioned above. The general Ramsey model does not generate such recursive blocks and neither the advanced method of steps implemented in Section 1 nor the theoretical arguments used in this section can be applied to this general model. Indeed, only a computational treatment is possible in the latter case.

Section 3: The general Ramsey vintage capital growth capital
To solve the general optimal growth model with CRRA utility functions, an elementary waveform relaxation is implemented as described in Magnus et alii (1997). The method works on the (discretized) structural optimization problem, and not on the first order optimality conditions. It roughly consists of a simple search process over the investment vector space associated with a straightforward waveform updating taking advantage of the structure of the state equations of the optimization problem. We check the accuracy of this method by numerically solving the linear utility case and comparing the obtained solution with the analytical solution derived in Section 2. Then, we check Benhabib and Rustichini's theoretical argument: for near-linear utility models, echoes effects are very strong and very persistent, although they vanish in the long run. Not surprisingly, the strenght and persistence of echoes effects decrease when the departure from the linear utility case increases. These numerical experiments, together with the numerical and theoretical results of Sections 1 and 2, give us a fairly good insight into how nonlinearities and echoes effects interact in vintage capital growth models.

Keywords: Vintage Capital Growth Models, Echoes Effects, Differential-Difference Equations, State Dependence, Method of Steps, Waveform Relaxation.

Scheduled for Session 5.5 Economic Growth