301 Econ F06 PROBLEM SET 2 - due in class on Monday Sept 25 Wissink 1. Critically evaluate the following statements and explain why or in what way are they true, false, or uncertain. a. Over the set of people in New York, the binary relation "is the half-brother of" is transitive. b. Over the set of people in New Jersey, the binary relation "is the mother of" is complete. c. If I am always willing to substitute six units of y for an additional unit of x, then my preferences violate the monotonicity axiom. 2. Draw indifference curve maps containing at least two indifference curves for the following descriptions of preferences and indicate the direction in which happiness is increasing. a. Ima Dancer loves to dance and hates housecleaning. She has strictly convex preferences. She does nothing else except dance and houseclean. She always is happier the more she dances. The more time she spends cleaning house, the less happy she is. b. Yogi always eats each hot dog he eats with exactly 2 oz. of mustard. Each hot dog eaten in this way provides 15 units of utility. Additional hotdogs without mustard, or mustard without hotdogs, provide no extra happiness (or harm) to Yogi. c. Given a choice between two bundles containing books and cassettes, Felicity prefers the bundle with the larger quantity of books. If the quantity of books is the same in the two bundles, Felicity is indifferent between them. d. Jake is a coffee addict; the more coffee he consumes, the more money he is willing to give up to obtain even more coffee. e. Ivanna consumes two goods: beer and pizza. She has a bliss point at 2 bottles of beer and 4 slices of pizza. Her happiness decreases as she moves farther and further (which one is it?) in any direction away from this bliss point. 3. Coach Steroid likes his players to be big, fast, and obedient. If player A is better than player B in two of these three characteristics, then Coach Steroid prefers A to B, but if B is better than A in two of these three characteristics, then Steroid prefers B to A. Otherwise, Steroid is indifferent between them. Wilbur Westinghouse weights 340 pounds, runs very slowly, and is fairly obedient. Harold Hotpoint weighs 240 pounds, runs very fast, and is very disobedient. Jerry Jacuzzi weighs 150 pounds, runs at average speed, and is extremely obedient. a. Does Coach Steroid have transitive preferences with respect to these three guys? b. After several losing seasons, Coach Steroid decides to change his way of judging players. According to his new preferences, Steroid prefers player A to player B if player A is better in all three of the characteristics that Steroid values, and he prefers B to A if player B is better at all three things. He is indifferent between players if they weigh the same, are equally fast, and are equally obedient. In all other cases, Coach Steroid simply says "the two players are not comparable." Are Coach Steroid's new preferences with complete respect to these three guys? c. Are Coach Steroid's new preferences transitive with respect to these three guys? How about over the population of all guys who might be on the team? 4. Al derives utility from 3 goods: music (M), wine (W), and cheese (C). His utility function is given by: u(M,W,C) = M + 2W + 3C. a. Assuming Al's consumption of music is fixed at 10, sketch the indifference curves between W and C for u = 40 and then again for u = 70. b. Show that Al's MRS of wine for cheese is constant for all values of W and C on the indifference curves calculated in part (a).
c. Suppose Al's consumption of music increases to 20. How would this change your answers to parts (a) and (b)? 5. Calculate the marginal utility of x, (pu/px), the marginal utility of y, (pu/py), and the MRS for each of the following utility functions. (Note: the p stands for a partial derivative.) u = 2x + 3y u = v(x) + y u = xa + yb 6. For each of the following statements interpret what the statement would imply about either the narrator's preferences and/or his utility function. That is to say, how might you represent the preferences of the narrator vis a vis the indifference curve map or the utility function or both. a. "It (RC Cola) is just as good as the high-priced brand (Coke)." b. "Peanut butter and jelly go together like a horse and a carriage." c. "Things go better with Coke." d. "Popcorn is addictive - the more you eat, the more you want." e. "Mosquitoes ruin a nice day at the beach." f. "A day without orange juice is like a day without sunshine." 7. Which of the following utility functions represent the same underlying preferences: a. u = x2y2 b. v = x y c. w = x + y
2 2 .5
u = 4x + 6y u = xy
u = ax + by u = xayb
u = 2(x.5) + y
u = ln(x) + y
u = (x+2)(y+1) u = (x+a)(y+b)
d. z = 10 + 2ln(x) + 2ln(y) e. g = 23 + xy f. h = 23 - xy
8. What kind of assumption(s) on preferences and/or utility functions would rule out the following: "bowed-out" indifference curves; crossing indifference curves; upward sloping indifference curves; kinked indifference curves; linear indifference curves. 9. Four university students, Abe, Betty, Charlie and Dave have been asked to assign utility numbers to five bundles containing various amounts of good x and good y, with more-preferred bundles getting higher numbers (see table below). These utility numbers will help us find a representation for each student's utility function UAbe, UBetty , UCharlie , UDave . Unfortunately, only one of the four people is a nicely behaved economics student with nicely behaved preferences that are consistent with the standard assumptions on consumer preferences. Which student is it? Can you suggest what is wrong with the others?
bundle E F G H I Quantities of good x good y 2 2 2 5 4 6 5 4 6 1 U Abe 2 4 6 6 4 Utility Index Numbers U Betty U Charlie U Dave 2 1 6 5 3 10 6 5 10 5 3 6 5 5 3
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