Calculus and Mean Value Theorem Based Optimal Consumption Decision Model Analysis: A Theoretical Framework

This study investigates the application of calculus-based optimization techniques and the Mean Value Theorem in constructing optimal consumption decision models under strict intertemporal budget constraints. The research develops a comprehensive theoretical framework that seamlessly integrates marginal utility analysis with the fundamental theorem of calculus to derive optimal consumption paths across multiple time periods. By utilizing the Lagrangian multiplier method alongside Euler-Lagrange conditions, the proposed model establishes the necessary mathematical conditions for achieving optimal consumption smoothing behavior over time. Furthermore, the analysis incorporates the Mean Value Theorem to rigorously prove the existence of optimal consumption points within continuous time intervals, demonstrating that consumption functions satisfying intertemporal optimality conditions must exhibit specific continuity and differentiability properties. To validate the framework, three theoretical case studies are systematically conducted: the classic two-period consumption optimization problem, the infinite-horizon Ramsey-type consumption model, and the complex consumption smoothing problem under deterministic income fluctuations. The results clearly indicate that calculus-based methods provide highly rigorous foundations for deriving optimal consumption rules, while the Mean Value Theorem offers essential analytical tools for proving both the existence and uniqueness of optimal solutions. Ultimately, this research significantly contributes to the theoretical literature on consumption behavior by formalizing the precise mathematical conditions under which optimal consumption decisions can be characterized. The findings have profound implications for advancing macroeconomic modeling, enhancing policy formulation, and improving long-term household financial planning strategies.