Optimal Allocation of Resources in an Open Economic System with Cobb–Douglas Production and Trade Balances

This paper develops a nonlinear optimization model for the optimal allocation of labor and investment resources in a three-sector open economy. The model is based on the Cobb–Douglas production function and incorporates sectoral interdependencies, capital depreciation, trade balances, and import quotas. The resource allocation problem is formalized as a constrained optimization task, solved analytically using the Lagrange multipliers method and numerically via the golden section search. The model is calibrated using real statistical data from Kazakhstan (2010–2022), an open resource-exporting economy. The results identify structural thresholds that define balanced growth conditions and resource-efficient configurations. Compared to existing studies, the proposed model uniquely integrates external trade constraints with analytical solvability, filling a methodological gap in the literature. The developed framework is suitable for medium-term planning under stable external conditions and enables sensitivity analysis under alternative scenarios such as sanctions or price shocks. Limitations include the assumption of stationarity and the absence of dynamic or stochastic features. Future research will focus on dynamic extensions and applications in other open economies.