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

1. Question 1. (25 marks) Suppose you want to minimize your expenditure on goods x and y having
prices p and q, respectively, subject to keeping your utility at a fixed level, U0. Your utility function
is given by U(x,y)= Bx1/2y1/2. (a) Write down the Lagrangian for this problem and find the first
order conditions. (b) Solve for the optimal values of x,y and the value of the Lagrange Multiplier.
(c) Interpret the Lagrange Multiplier, . (d) Verify the second order condition is satisfied. Are your
solutions a minimum or a maximum? (e) Find the value function. (In this case it is called an
expenditure function.)