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3. Demand and Comparative Statics (67 points) 1. (22 points) It is the year 2042, and your preferences over robot butlers and Soylent Green are given by U = Zinxrb + 7inxsg, while your budget constraint takes the form 311 = prbxrb + psg xsg - You remember the world in 2021, covered in trees and beaches, and how so many others told you the xture would be a utopia free from scarcity. However, because you studied economics, you knew there would be benets and costs as you lurched helplessly into the future, and you knew that scarcity would remain. Looking out from your house boat as it rocks on toxic seas, you think that robot butlers are really freaking cool... Let's examine the work done in concert between your brain chip and Amazon's algorithm as they perfectly consume for you on a particular day, given current income and prices. a. (3 points) Mathematically derive the demand mctions for xrb and x59. 13. (3 points) Are the demand functions downward sloping? That is, calculate the own-price effect for each good and use that result to explain for each good if it is an ordinary good or a Gifn good. c. (3 points) Derive the income effect for each good. Are they normal or inferior goods and why? d. (3 points) Calculate the cross-price effect for both goods. Are the commodities substitutes, complements, or unrelated? Use mathematics and intuition to support your answer. e. (2 points) Find it. That is, take an appropriate derivative to calculate the marginal utility of income. Comment on the meaning of your result as it relates to your life and consumption. f. (4 points) Let the initial income be m = 30,000, p\