4. The Buffet You have decided to eat sushi for lunch and have a grand total of $20 to spend. Whatever money you have left over from buying sushi rolls will be spent on cans of soda to get you through the rest of the day. Your problem is to choose the best combination of sushi rolls and cans of soda.

(a) You decide to head over to Oshio Cafe due to its convenient proximity to the economics department. A sushi roll is $6 and cans of soda are $1. Sketch your Problem Set 1 3 budget constraint, being certain to label any relevant intercepts. ( Throughout this problem, assume that you can purchase fractions of sushi rolls and fractions of cans of soda.)

(b) Let's say that your utility for a bundle of sushi rolls (R) and cans of soda (C) is given by the following function: U(R, C) = 30R 1 3 C 2 3 . (2) Derive the marginal utility of R and the marginal utility of C.

(c) What combination of sushi and soda maximizes your utility given your budget constraint? Add this optimal point and the indifference curve passing through it to your graph of the budget constraint.

(d) Now suppose that you decided to go to the buffet at Fuji Chef instead. The buffet costs $13. Sketch your new budget constraint. (Hint: your possible consumption bundles still consist of cans of soda and rolls of sushi.) (e) Explain in words what happens to the number of cans of soda and sushi rolls in your optimal bundle, given the utility function in part (b).

(f) Fuji Chef still has sushi left at the end of lunchtime. If we assume that nobody was rushed and everybody got a chance to eat as much sushi as they wanted, something must be wrong with our utility function. After some level of sushi, it must be that more sushi actually makes us worse off. This level of sushi is considered a satiation point. More sushi is good up to this satiation point, the point at which we happily say "I'm full," but bad after this point, when we say instead, "I shouldn't have eaten so much." Sketch one more set of indifference curves where more soda is always better but we have a satiation point for sushi after six rolls. Sketch three or four indifference curves and indicate the direction of increasing utility.

(g) Do the preferences in part (f) satisfy all of our properties of well-behaved preferences (complete, reflexive, transitive, monotonic, convex)? If any of the properties do not hold, explain why not.