9. Consider a consumer who chooses a bundle of goods to maximize utility, subject to an affordability constraint:

(, ) = +

= (sub) + (sub)

= 100; sub = 1; sub = 4

a. Write down the Lagrangian function, = (, , ), for this problem and differentiate it with respect to x, y, and .

b. Solve the first order necessary conditions for the utility maximizing quantities of good X and good Y: x* and y*.

c. What is the Lagrange multiplier (*) associated with the utility-maximizing bundle?

d. Use the Hessian matrix of the Lagrangian function's second derivatives, H(L), to verify that you have, indeed, identified a constrained maximum point.