9. My grandmother, Rosie, only cared about two goods, cake and Diet Coke. Suppose she had a fixed income of M dollars per week, and the prices of cake and Diet Coke were, respectively, p and q. Her utility function had a "Cobb-Douglas" form: U(H, B) = H1/BB2/3, where H represents the number of ounces of cake she consumed in a given week and B represents the number of bottles of Diet Coke she consumed. It follows that if she had market basket (H, B), her marginal utilities of cake and Diet Coke were MUH = = (B/H) 2/3 and MUB = =(B/H)-1/3. a. Does this utility function indicate that Rosie received diminishing marginal utility for both goods? In other words, did she receive a smaller amount of marginal utility for an ounce of cake when she was already consuming a larger amount of cake? Was that true of Diet Coke too? b. What is her marginal rate of substitution if she currently has H ounces of cake and B bottles of Diet Coke? Interpret your answer--what is the minimum number of ounces of cake you would have to give her to convince her to trade you one bottle of Diet Coke? c. Write down an equation for Rosie's budget constraint. d. Find mathematical expressions for Rosie's demand function for cake. e. What is her elasticity of demand for cake? (Use your result in part c. Hint: The easiest way to compute it here may be to use changes in logarithms, i.e., (Alog H /A logp), holding everything else constant.) Does it depend on how much cake or pickles she is consuming? f. What is her cross-elasticity of demand for cake with respect to the price of Diet Coke? Does it depend on the market basket she is consuming? g. What is her income elasticity of demand for cake? Does it depend on her income? What prices, if any, does it depend on