Output Maximization Subject to a Nonlinear Constraint

The main aim of this paper is to derive the mathematical formulation to device an optimal purchasing policy for the service providing agency. An attempt has been made to maximize the output function of an agency subject to a nonlinear budget constraint by assuming that the agency gets price discounts for purchasing larger quantities of other inputs. Such quantity discounts alter the linear budget constraint and result in a nonlinear (convex type) budget constraint. We use the method of Lagrange multipliers and apply the firstorder necessary conditions as well as the second-order sufficient conditions for maximization. We also use comparative static analysis and study the behavior of the agency when prices of inputs undergo change, besides providing useful interpretation of the Lagrange multipliers in this specific case. Illustrating an explicit example, we show that the optimization problems play an important role in the real world.