please answer fully.

Problem 4 From the Optimal Choice to Demand This exercises builds on Problem 3 from Problem Set 2. In Problem Set 2 you used the Lagrangian method to find a consumer's optimal choice. In this exercise, you go one step further and find the consumer's demand for food (x). 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) = x1/3y2/3. Carla has a monthly budget of $900 and currently clothes cost $25 per item. a) Call Px the price of food. Write the expression of Carla's budget constraint leaving the price of food as a parameter Px. b) Write Carla's utility maximization problem. c) Now, using the Lagrangian method find Carla's optimal bundle keeping the price of X represented by the parameter Px. c.1) Write the expression of Carla's Lagrangian function. c.2) Take the derivatives respect to x, to y, and to A (first order conditions of the Lagrangian problem) c.3) Solve for the endogenous variables x, y, and ). Keep in mind that they might be functions of the parameter Px. d) In a graph, measuring food along the horizontal axis and the price of food along the vertical axis illustrate Carla's demand curve. Is it downward sloping? e) A good X is related to good Y if a change in the price of X affects the quantity of Y the consumer wishes to buy. In this exercise, does Carla consider food a related good to clothes