Abstract
Research in macroeconomic growth remains extremely relevant and important in the modern world. Macroeconomic growth focuses on analysing the dynamics of production, income, and consumption, enabling the development of scenarios for long-term development. The technology of using the theory of optimal control for modelling single-product growth macroeconomics plays an important role in understanding and forecasting economic development. The article proposes a technology for using the theory of optimal control to model a single-product growth macroeconomy. Technological tasks for modelling single-product macroeconomic growth are formulated, which can be initial (at the beginning of the production process), intermediate throughout the entire duration of the process, and final at the end state of production. In order to construct a model of optimal macroeconomic growth, several assumptions are taken into account. Specifically, gross output is decomposed into final output and production consumption. Production consumption is assumed to be proportional to gross output. Final output is further divided into non-production consumption, total investment, government expenditure, taxation, pollution abatement, and balance of trade (exports minus imports). To build the optimal process, where consumption rate acts as the control variable and specific capital represents the phase trajectory, a target function (goal criterion) is formalized. The target function aims to maximize the average (integral) discounted consumption over a specified time interval. Sufficient optimality conditions are used to investigate the model of optimal control. The Lagrange method is employed to account for the constraint on minimum consumption. This allows transforming the constrained optimization problem into an unconstrained optimization problem involving the optimization of two functions with multiple variables. During the investigation of the single-product macroeconomic growth model, an algorithm has been developed to calculate the optimal process for three formulated technological tasks under selected three-tier regimes. Among the built optimal processes, it is possible to select a priority optimal process.
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