Using a discrete version of the Ramsey Vintage Capital Model we provide a characterization of the initial capital stocks compatible with a predefined scrapping time and a level of technical progress which generate feasible capital paths. From that characterization, it is proved that for a given level of technological progress there exists a minimum scrapping time for the machines which allows a sustainable growth path. Moreover, we reduce the infinite horizon dynamic programming problem to one of finite dimension and we show how this reduction allows us to find the optimal structure of the initial capital stock as well as the optimal scrapping time. A numerical example shows that, in accordance with the infinite horizon approach, the greater the technological progress, the lower the optimal scrapping time.