Analytical And Numerical Methods For The Analysis Of Vintage Capital Models In Continuous Time

In this paper, we propose a general analytical and computational framework to handle a large class of vintage capital models in continuous time. Concretely, our treatment applies to vintage capital models with clay-clay and putty-clay technologies, which covers most of the models recently published in the macroeconomic literature. We build on the relaxation algorithm developped in Boucekkine et al. (1999).i) We first come to analyze very precisely all the steps of the latter algorithm. In particular, its integration and cyclic coordinate descent procedures are discussed, mainly in terms of robustness. To this end, some alternative algorithmic schemes are examined. For comparison purposes, we solve the nonlinear Ramsey vintage capital model as dealt with in Boucekkine et al. (1999).ii) Second, we try to provide an assessment of the accuracy of the method by solving an optimal growth model with capital maintenance costs which admits some closed-form solutions.iii) We finally present some analytical tools aimed at complementing the computational experiments. These tools are shown to be of high interest as far as a rigorous investigation of the stability of the models is required. The problem involved is usually the computation of the roots of a transcendental function. We point at some useful procedures to detect the most "important" roots in terms of stability.