1.Suppose Anne's utility function for food (X) and clothing (Y) is U(X, Y) = 4X1/2 + Y. She faces prices Px, Py and income I. a. Assume an interior solution and using the Lagrangian method, find Anne's demand functions for X and Y. Show all your work. (Hint: notice that when you take the ratio of the first 2 FOCs, you will find X only as a function of exogenous variables, that is you will find the demand for X at that step. Then substitute this into the 3rd FOC to derive the demand for Y). b. Find Anne's optimal bundle (B) if Px = 1 and Py = 1 and she has I = 24. Show bundle B on a ICBL diagram. c. Find the optimal bundle (C) if Px and I remain the same, but Py increases to 8. Show bundle C on the diagram above. d. Compare the MRS and slope of the budget line at bundle C.

2. In Question-1 where, Anne's utility function for food (X) and clothing (Y) is given as U(X, Y) = 4X1/2 + Y, for interior solutions a. show mathematically how Anne's demand for X depends on her income. b. Draw Anne's Income Expansion Path and Engel Curve for good X. c. Calculate the cross-price elasticity of demand for X.

3. Martha's preferences are given as U = XY. a. Using the Lagrangian method, find the compensated demand functions for X and Y. b. At prices Px = 20, and Py = 10, what is the minimum amount of income necessary for Martha to attain a level of utility, U = 71.86? c. Show your result in (b) on an IC-BL diagram.