This chapter develops necessary conditions for optimality in constrained optimization problems. Building on classical Lagrange theory, it introduces the Kuhn–Tucker conditions as a generalization applicable to inequality constraints. The chapter explores the rationale of the Kuhn–Tucker conditions and their relationship to a saddle point of the Lagrange function. It illustrates the main ideas through numerical examples, and shows their usefulness in qualitative economic analysis. Applications such as peak-load pricing, revenue maximization under various constraints, and the effects of different instruments of environmental regulation demonstrate how optimization techniques lead to powerful economic insights.