This document describes an optimization problem where an individual aims to minimize expenditure on goods x and y subject to maintaining a fixed utility level U0, given prices p and q for each good. The utility function is U(x,y)= Bx^1/2y^1/2. The Lagrangian is written down and first order conditions are found by taking partial derivatives with respect to x, y, and the Lagrange multiplier. The optimal values of x and y are then solved for, along with the value of the Lagrange multiplier. The economic interpretation of the Lagrange multiplier is provided. It is verified that the second order condition is satisfied, indicating a minimum is found. Finally, the expenditure function giving the minimum expenditure