Joan has preferences over two goods, x and y, that admit the following utility function represen- tation: U(x, y) = xy Joan's constrained optimization problem is: maxx,y U(x, y) = xy subject to px x + py y = I where px and py are the respective prices of goods x and y and I is income. (a) Derive Joan's demand functions for goods. Your answer should show that quantity de- manded in a function of income and own price. 3 points (b) Suppose that px = py = 1 and that I = 8. What is Joan's optimal bundle? 1.5 points (c) Now suppose that the price of good x doubles to px = 2 while everything else stays con- stant. What is Joan's new optimal bundle? 1.5 points (d) Draw a graph that shows how Joan's behavior changes when the price of x doubles. This graph should show how Joan's changing consumption can be decomposed into income and substitution effects. Please label everything relevant to the problem, including the coordi- nates of the original optimal bundle, the "substitution effect" bundle, and the new optimal bundle. 4 points Note: For a 1/2 point penalty, you may omit calculating the substitution effect bundle. Bonus: Calculate the amount of additional income necessary (Hicksian compensation) necessary to keep Joan on her original indifference curve post price change.