About additional financing in the "investment-consumption'' model

In investment-consumption models, the growth rate of capital-labor ratio is the difference between own investments and the rate of amortization. Let the goal of maximizing the reduction of production time at a given level of capital-labor ratio be set and, to achieve this goal, additional attractive funds of a given volume, which, however, are involved in the process in the form of a financial flow of a limited highest level - the maximum possibility of distributing investments (it is assumed that the specified value represents a smooth financial capital-labor ratio). The functioning of the "investment-consumption'' model with a fixed share of investment in the produced value is considered. The process involves additional investments of a total volume S, which arrive in the form of a financial flow, and the rate of receipt of funds is limited from above by the value of the maximum ability to absorb investments. The question of the form of financial flow that will ensure reaching the required level of capital ratio in the shortest possible time is being studied. It turns out that the flow we are looking for is structured as follows. There is a pair of capital-labor ratio values between the initial and target values, such that while the capital-labor ratio changes from a lower to a larger value, only own investments are used. The rest of the time, additional funds are used at the maximum possible rate. Formulas for calculating the indicated values of capital-labor ratio are obtained.