3. Let U(X,Y) = xy. a. Write out the optimization model and Langrangian for the expenditure minimization problem. Be sure to label MRS and the expansion path. Note, the second-order condition for this problem was satisfied! b. Derive the first-order conditions and solve for X and Y, the expenditure-minimizing bundle. c. Substitute X and Y into the direct expenditure function and interpret the resulting expression. d. Take the partial derivatives of your expression from 3c with respect to P, and P,. Interpret the resulting expressions. 4. Verify slope and symmetry requirements for the Hicksian demands from 3b. Verify cross- price symmetry of the Hicksian demands.