Problem 3 The Optimal Bundle {When the consumer spends money on all items) Suppose Carla spends her money on food (x) and clothes (y). Carla's tastes for these two goods are described by the utility function U(x,y) = xT/3y2/3. Suppose food costs $5 per meal and clothes cost $25 per item. a) How does Carla feel about food and clothes? Does she prefer consuming them in even or uneven bundles? b) What happens to Carla's marginal utility for food as she consumes more of it? Does it increase, stay the same, or decline? c) What is the expression of Carla's budget constraint when her monthly budget is $900? d) How can you formally describe Carla's utility maximization problem? e) Find Carla's optimal bundle using the Lagrangian method. e.1) Write the expression of Carla's Lagrangian function. e2) Take the derivatives respect to x, to y, and to l. (first order conditions of the Lagrangian problem) e3) Solve for the endogenous variables x, y, and 1.. f) In a graph, measuring food along the horizontal axis and clothes along the vertical axis illustrate Carla's optimal bundle. Carefully label all intercepts and curves as well as the optimal amount of food and clothes in the optimal bundle. g) How much money does Carla spend on food if she purchases her optimal bundle? How much money does she spend on clothes? h) Do you notice a relationship between Carla's food budget share and the exponent assigned to food in Carla's utility function? Do you notice a relationship bemen Carla's clothes budget share and the exponent assigned to clothes in Carla's utility function? 11) How can you test if this relationship holds for any level of budget and is not driven by Carla's having exactly $900 to spend