1. Oliver is a graduate student who receives utility from the power cooling his home down to 75 (x1) and the power he uses for his own entertainment at night (x2). He has utility from consumption of each equal to: U(x, x) = 4x75x2.25. Oliver has income I>0, and faces energy prices P> Pa>0, where the subscript "n" stands for night and "d' stands for day. Assume that Oliver will consume strictly positive amounts of both x1 and x2. Oliver may chooses to cool down his house during the day time (so called "precooling"), or during the night when he gets back home. If he chooses precooling, there would be a "leakage" of x1 at 20%, that would not contribute to his utility. b) a) Set up the constrained utility maximization problem for Oliver if he chooses precooling. Solve Oliver's utility maximization problem using the Lagrangean approach to find Oliver's Marshallian demand functions. c) d) e) f) g) Derive the indirect utility function V (p; I) using your results from (b) and verify that Roy's identity holds. Set up the constrained utility maximization problem for Oliver if he chooses cooling at night. Solve Oliver's utility maximization problem using the Lagrangean approach to find Oliver's Marshallian demand functions. Under what condition will Oliver choose one way over the other? Since he is a poor graduate student, Oliver places more emphasis on minimizing his costs of energy use, while ensuring his utility is at least U. Set up Oliver's expenditure minimization problem when he chooses precooling. h) Solve Oliver's expenditure minimization problem using the Lagrangean approach to find Oliver's Hicksian demand functions. i) Derive the expenditure function E(p; U) using your results from (e) and verify that Shepard's lemma holds. j) Due to the cost of communicating prices, suppose Oliver faces with constant energy price over time at the weighted average po, where Pn> Po > Pa >0. How would your answers to parts a), d), f), and g) above change?