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301 Econ F06 ANSWERS PROBLEM SET 2 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. False, the relation "is the half-brother of" is not necessarily transitive. Here's an example. Suppose Joe and Mary beget Joey; Mary and Fred beget Freddy; and Fred and Jackie beget Jack. Then, Joey is the brother of Freddy and Freddy is the brother of Jack; but Joey is NOT the brother of Jack. b. Over the set of people in New Jersey, the binary relation "is the mother of" is complete. False, the relation "is the mother of" is not complete over. For example suppose you and I both come from NJ. I am not your mother and you are not my mother and we are not each other's mothers, so, the relation is not 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. No, they violate the STRICT CONVEXITY assumption. Can't refer to the notion of monotonicity at all here since you are getting more of one good and less of the other. 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. See last page. 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? Coach Steroid does not have transitive preferences, since he prefers Jerry to Wilbur even though he prefers Wilbur to Harold and he prefers Harold to Jerry. 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 complete with respect to these three guys? Coach Steroid's new preferences are not complete because with these new preferences he cannot compare Wilbur and Harold. 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? Over these three guys, the coach's preferences are transitive by default. Since you can not prove they are not transitive, then they are. But more importantly, Coach Steroid's new preferences are transitive. If there are any three players, say Jack, Ryan, and William, such that Coach Steroid prefers Jack to Ryan and Ryan to William, under these new preferences, then it also has to be the case that Coach Steroid prefers Jack to William. 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. See last page. 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). MRS = MUc/MUw = 3/2 c. Suppose Al's consumption of music increases to 20. How would this change your answers to parts (a) and (b)? The quantities of C and W yielding u = 40 and u = 70 would be smaller, but the new indifference curves corresponding to u = 40 and u = 70 would have the same slope as the previous ones. The MRS between C and W is a constant. (This situation is similar to one with only two goods and a monotonic transformation of the utility function. The indifference curves with the original function are still indifference for curves the transformed function; only the specific utility units attached to the indifference curves change.) 5. Assume "x" is the horizontal good and "y" the vertical good. u = 2x + 3y => u = 4x + 6y => u = ax + by => u = 2(x.5) + y => u = ln(x) + y => u = v(x) + y => u = xy => u = xayb => u = (x+2)(y+1)=> u = (x+a)(y+b) => u = xa + yb => MUx = 2; MUx = 4; MUx = a; MUx = 1/x.5; MUx = 1/x; MUx = v'(x); MUx = y; MUy = 3; MUy = 6; MUy = b; MUy = 1; MUy = 1; MUy = 1; MUy = x; MRS = 2/3 MRS = 2/3 MRS = a/b MRS = 1/x.5 MRS = 1/x MRS = v'(x) MRS = y/x MUx=axa-1yb ; MUy=bxayb-1 ; MRS=ay/bx MUx = (y+1); MUx = (y+b); MUx = axa-1; MUy = (x+2); MUy = (x+a); MUy = byb-1; MRS = (y+1)/(x+2) MRS = (y+b)/(x+a) MRS = (axa-1)/(byb-1) 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)." RC and Coke are perfect substitutes. 2 b. "Peanut butter and jelly go together like a horse and a carriage." Peanut butter and jelly are perfect complements. c. "Things go better with Coke." The marginal utility of all other goods is increasing in the level of Coke. (An example of a utility function with this property is u(x,y) = xy + y, where y is Coke and x is some other good.) d. "Popcorn is addictive - the more you eat, the more you want." The narrator has an increasing MRS between popcorn and "all other goods." e. "Mosquitoes ruin a nice day at the beach." Mosquitoes are a "bad." f. "A day without orange juice is like a day without sunshine." Let y = OJ and x = anything else. Then u(x,0) = 0 for all x 0 and u(x,y) > 0 for all x 0 and y>0. (Note, u(x,y) = xy + y would work here too.) 7. Which of the following utility functions represent the same underlying preferences: Functions a, d, and e represent the same preferences. a. u = x2y2 b. v = x y c. w = x + y 2 2 .5 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. Convex preferences rule out bowed-out indifference curves; transitivity and monotonicity together will rule out crossing indifference curves; monotonicity rules out upward sloping indifference curves; differentiability rules out kinked indifference curves; strict convexity rules out 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? For each person, plot the bundles and then look at the indifference curves that would be implied by the utility numbers, noting that if the bundles receive the same utility number they are on the same indifference curve. Abe is just fine. Betty appears to violate the notion of convex preferences. Charlie's indifference curves cross, so his preferences either violate monotonicity or transitivity. Dave's preferences violate the monotonicity assumption. 3 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 ABE 7 6 5 4 3 2 1 0 0 2 4 6 8 Betty 7 6 5 4 3 2 1 0 0 2 4 6 8 Charlie 7 6 5 4 3 2 1 0 0 2 4 6 8 Dave 7 6 5 4 3 2 1 0 0 2 4 6 8 4 2a. 2b. IC dnig acn 1 IC 0 dogs IC 0 IC 1 2 1 cleaning 2 4 mustard 2c. books 2d. $aog IC IC 1 0 IC 0 IC 1 cassettes coffee 2e. 4a. beer wine 30 2 Q.4a 15 IC 0 4 pizza IC 1 10 20 cheese 5
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Path: Wisconsin >> ECON >> 101 Spring, 2007
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Path: Wisconsin >> ECON >> 101 Spring, 2007
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Path: Wisconsin >> ECON >> 101 Spring, 2007
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Path: UT Dallas >> ECON >> 2302 Spring, 2006
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Path: UT Dallas >> ECON >> 2302 Spring, 2006
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Path: Wisconsin >> ECON >> 101 Spring, 2007
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Path: Carnegie Mellon >> STAT >> 220 Fall, 2005
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Path: UT Dallas >> ECON >> 2302 Spring, 2006
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Path: UBC >> CPSC >> 303 Winter, 2007
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Path: UBC >> CPSC >> 303 Winter, 2007
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Path: UBC >> CPSC >> 303 Winter, 2007
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Path: UBC >> CPSC >> 303 Winter, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
Description: ...
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Path: JMU >> ECON >> 201 Fall, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Syllabus
Path: UCSB >> ECON >> 01 Summer, 2007
Description: Econ 1: Principles of Microeconomics I. Administrative Information Lecturer: Francisco Azeredo Office: NH 2028 Teaching assistant: Ben Hansen Office: NH 2044 Class: MTW TA section: R R 9:30 10:55am 9:30 10:55am 12:30 1:55 pm Office Hours: TW 11:00...
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Path: UCSB >> ECON >> 01 Summer, 2007
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chapter 4 answers
Path: JMU >> ECON >> 201 Fall, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Path: Wisconsin >> ECON >> 101 Spring, 2007
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Path: JMU >> ECON >> 201 Fall, 2007
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Homework2
Path: UCSB >> ECON >> 01 Summer, 2007
Description: UNIVERSITY OF CALIFORNIA SANTA BARBARA Economics 1 Principles of Economics Summer 2007 Session A Homework #2 I. Problems: Chapter 3 I.1 Book Problems Francisco Azeredo Ben Hansen Please place your answers in the space below. Take the time to draw ...
