(401) An extension of Blach's optimum savings model. امتداد لنموذج بلاخ الأمثل للادخار - A. qayum (The institute of national planning., 2018)
This paper, entitled “An Extension of Black's Optimum Savings Model,” attempts to expand John Black’s model of optimal savings, which itself was based on Ramsey’s theory of savings. The author argues that Black’s framework is limited because it assumes that capital alone is responsible for generating income. Such an assumption is unrealistic, as actual economic systems rely on the joint contribution of both labor and capital. Therefore, the study seeks to develop a broader analytical framework that incorporates these two productive factors simultaneously. The analysis assumes an economy with fixed production techniques, implying a constant output-capital ratio, and also assumes the existence of a constant optimal savings rate. Starting from the principle of utility maximization, the paper argues that the utility derived from present consumption should be equal to the discounted stream of future utilities generated by savings over time. The study employs a marginal utility function and a constant time-discounting rate to derive mathematical expressions for determining the optimal savings rate. The findings indicate that the optimal savings rate depends not only on the time-discounting rate but also on the output-capital ratio and the utility parameter. This conclusion differs from Black’s earlier argument, which suggested that under conditions of zero time discounting, the optimal savings rate becomes independent of capital productivity. The paper further extends the analysis to include both gross and net savings and derives equations for determining their optimal levels. The study is significant because it bridges the conceptual gap between the rate of return on capital and the output-capital ratio, thereby providing a more realistic analytical framework for understanding savings and investment behavior in economic systems.