A Dynamic Optimization Model Considering Cost and Speed in the Cobb-Douglas Production Function

This paper addresses the problem of cost minimization in a dynamic production environment where total costs depend not only on the production amount but also on production speed—a factor often overlooked in traditional production models. Unlike previous studies that focus solely on production volume, we incorporate production speed as a critical component, reflecting real-world scenarios where faster operations lead to additional costs such as machinery wear, energy consumption and training cost for workers. Utilizing the Cobb–Douglas production function, we derive a comprehensive total cost function and introduce an innovative application of the Euler–Lagrange equation to minimize these costs. This approach, rarely applied in the context of optimal production planning, enables us to determine the time-dependent optimal production quantity. Our analysis reveals that the optimal production amount evolves in an increasing and concave pattern over time, offering a novel and practical framework for companies seeking to balance production amount and speed for cost-efficient management. • Apply the Cobb–Douglas production function to derive Cost function depending on production speed. • We assume that the production speed is additively separable in the total cost. • Use the Euler–Lagrange equation to solve the cost minimization problem. • Obtain the optimal production quantity. • Shows the optimal production quantity is increasing and concave in time.