Feasibility and optimality of the initial capital stock in the Ramsey vintage capital model

Using a discrete-time version of the Ramsey Vintage Capital Model we provide a characterization of the set of initial capital stocks compatible with a predefined scrapping time, given the rate of technical progress and the level of capital productivity. Each profile of initial capital stock in that set generates a complete infinite horizon feasible capital path. From that characterization, we prove the existence of a minimum value for the scrapping time of the machines compatible with the rate of technological progress. Moreover, for each level of capital productivity, there exists an upper bound for the technological progress which allows the existence of feasible capital paths with full employment. Finally, we transform the infinite horizon dynamic programming problem into one of finite dimension. We use this to find the optimal lifetime for the machines as well as the optimal composition of the initial capital stocks. A numerical example shows that, in accordance with the infinite horizon approach to the problem, the increase in the rate of technological progress leads to a decrease in optimal scrapping time of capital goods.