Econ 301 – F07
ANSWERS to PROBLEM SET 2 - due in class on Thursday September 20
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 State, the binary relation "is the Facebook Friend of" is complete and transitive.
[ANSWER] False, the binary relation “is the Facebook Friend of” is not complete and not transitive .
Completeness: For example, take Jane and Mike who both live in NY and are not Facebook Friends
(FBF). So Jane is NOT Mike’s FBF and Mike is not Jane’s FBF, or both, so the relation is NOT complete. Transitivity: for example, Jane is the Facebook Friend of Mike. Mike is the Facebook
Friend of Carol. To be transitive, Jane MUST BE the Facebook Friend of Carol. But, in reality,
Jane may not be the Facebook Friend of Carol. b.
Assuming a 2 good commodity space and an agent with well behaved preferences, the binary relation “is strictly preferred to” is complete.
[ANSWER] False, the binary relation “is strictly preferred to” is not complete . Suppose Bundle A and Bundle B are indifferent to each other. Then A is not strictly preferred to B and B is not strictly preferred to A, or both, so the relation “is strictly preferred to” it not complete.
c.
Unlike Prof. Wissink, an indifference curve can never be “thick.”
[ANSWER] True, a well behaved indifference curve can never be “thick.” A thick indifference curve violates the monotonicity assumption, which means that more is better. Suppose that (x1, y1) is a bundle of goods and (x2, y2) is a bundle of goods with strictly more of both goods x and y. By the monotonicity assumption, (x2, y2) is strictly preferred to (x1, y1). But if two bundles are on the same thick indifference curve, (x2, y2) is indifferent to (x1, y1). y
(x1,y1)
(x2,y2)
IC x

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. (Note, not all these individuals will have nicely behaved preference, in fact, maybe none of them do.) a.
Ima Slacker loves to go shopping for “all other goods” and dislikes working. She has strictly convex preferences between “all other goods” and working. b.
James Bond’s favorite drink is a Vodka Martini, shaken, not stirred. His personal perfect recipe is to always use 1 part vodka to 3 parts gin. c.
Jane describes nirvana as exactly 1 piece of cheesecake with ¼ cup of raspberry sauce. d.
Mikko cares about road quality in Oulu Finalnd and could care less about road quality in Ithaca New
York.
[ANSWER]
(a) (b)
IC1
All other goods Gin (part)
IC0
3
IC1
IC0
Working
0
1
(c) (d)
Raspberry (cup)
IC1
1/4
IC0
Ithaca
IC0
0
0
1
Cheesecake (piece)
IC1 IC2
Vodka (part)
Oulu
2

3.
Al derives utility from 3 goods: music (M), wine (W), and cheese (C). His utility function is given by: u(M,W,C) = MWC a.
Assuming Al's consumption of music is fixed at 10, sketch the indifference curves between W and C for u = 400 and then again for u = 700 putting W on the horizontal and C on the vertical.
[ANSWER] So 400=10WC

C=40/W is the equation for the indifference at u=400 and similarly,
C=70/W is the equation for the indifference at u=700.
Indiffe re nce Curve s be twe e n W and C
C
8
6
4
IC1 (u=700)
IC0 (u=400)
2
0
0 20 40 60 80
W b.
Show that Al's MRS between wine and cheese falls as Al moves down and to the right along his indifference curve.
[ANSWER] There are several ways you can prove this. The easiest in this case is to note that the equation for the indifference curve is C=K/W where K is some constant depending on the level of utility chosen as fixed. Now note that dC/dW = -K/W
2
and that d
2
C/dW
2
= 2K/W
3 which is positive, which means that the negative slope of the indifference curve is getting algebraically LESS negative, i.e., flatter, which, in absolute value, means that the MRS is falling. c.
Suppose Al's consumption of music increases to 20. How would this change your answers to parts
(a) and (b)?
[ANSWER]
(a) part: New IC0 and IC1 shrink. Since Al consumes twice music, he consumes less wine and cheese to keep the same utility.
Indiffe re nce Curve s be twe e n W and C
IC1
60
IC0
40 ne wIC1
20 ne wIC0
0
0 2 4 6 8
W
3

(b) part: Al’s utility function when M=20, u = 20WC. Then, MUw = 20C and MUc = 20W. Al’s
MRS = C/W. Al’s MRS is the same as in (b). There is no change in the answer.
4.
Calculate the marginal utility of x, ( p u/ p x), the marginal utility of y, ( p u/ p y), and the MRS for each of the following utility functions. (Note: the p stands for a partial derivative.) a. u = 2x + 3y c. u = 4x + 6y e. u = ax + by g. u = 2(x
.5
) + y i. u = ln(x) + y k..u = x a
+ y b b. u = v(x) + y d. u = xy
[ANSWER] a. u = 2x + 3y b. u = v(x) + y
MUx
2 p v(x)/ p x c. u = 4x + 6y d. u = xy
4 y e. u = ax + by f. u = x a y b g. u = 2(x
.5
) + y a ax a-1 y b x
-0.5
h. u = (x+2)(y+1) y+1 i. u = ln(x) + y 1/x j. u = (x+a)(y+b) y+b k. u = x a
+ y b ax a-1 f. u = x a y b h. u = (x+2)(y+1) j. u = (x+a)(y+b)
MUy
3
1
6 x b bx a y b-1
1 x+2
1 x+a by b-1
MRS
2/3 p v(x)/
2/3 y/x a/b p x
(ay)/(bx) x
-0.5
(y+1)/(x+2)
1/x
(y+b)/(x+a)
(ax a-1
)/(by b-1
)
5.
Which of the following utility functions represent the same underlying preferences: u = 2x + 3y v =
℮ (2x + 3y) w = x
2
+ y
3 z = 10 + 2ln(x) + 3ln(y) g = 23 + 4x + 6y h = -23 - 2x – 3y
[ANSWER] The utility functions, u, v, and g represent the same preferences. Monotonic transformations of the utility function, u, are v and g. Notice that the function, h has the same MRS as u.
But h is not a monotonic transformation of u.
Utility functions
(A) u = 2x + 3y
A transformation of (A)
(B) p (B)/
(C) p u MUx
2
MUy
3
MRS
2/3 v =
℮ (2x + 3y) w = x
2
+ y
3 v =
℮ u v` =
℮ u
> 0 2
℮
2x
(2x + 3y)
3
℮ (2x + 3y)
3y
2
2/3
(2x)/(3y
2
) z = 10 + 2ln(x) + 3ln(y) 2/x 3/y (2y)/(2x)
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g = 23 + 4x + 6y h = -23 - 2x – 3y g = 23 + 2u h = -23 - u g` = 2 > 0 h` = -1 <0
6.
Suppose using empirical evidence that you discover that a good approximation of the average U.S. consumer’s indifference curve map over the two goods Clothing per year and Food per year is as illustrated below. What might you suggest is a
Food at home per year reasonable interpretation for such an indifference curve map?
4
-2
6
-3
2/3
2/3
[ANSWER]
IC_2
IC_1
Clothing per year
IC_3
One interpretation of this indifference curve map is that there are minimum levels of food and clothing necessary to support life. The consumer cannot trade one good for the other if it means having less than these critical levels. As the consumer obtains more of both goods, however, the consumer is increasingly willing to trade between the two goods and they start looking more and more like perfect substitutes for one another.
7.
The Wissink kids, Gerrit, Christine and Greg are trying to decide where to go to dinner one night.
There are three choices: Hals Deli, Subway and Taco Bell. Gerrit says his ranking of the three, from best to worst is: Hals, Subway and then Taco Bell. Greg's preferences are: Subway, Taco Bell, and then Hals. Christine's preferences are: Taco Bell, Hals and then Subway. a.
Gerrit proposes they take each pair of alternatives and let a majority vote determine the family preference ranking over each pair of alternatives. That is to say if “A” goes up against “B”, the alternative with more votes is preferred to the other. Determine the family ranking over each pair of alternatives based on Gerrit's proposal. Over the three alternative eateries, is this family’s ranking complete? Is this family’s ranking transitive?
[ANSWER] Using majority voting over each pair of places we get: TB vs. H results in TB winning
(Christine and Greg vote for TB over H). So: TB is preferred to H. Now consider H vs. S and you get: H is preferred to S. Now consider TB vs. S and you get that: S is preferred to TB. So the family's preferences just cycle around and around with no un-dominated choice emerging (under this method of majority rule). The family's ranking, however , is complete since we can always compare any two eateries this way. The family's ranking IS NOT transitive since TB is preferred to H and H is preferred to S but it is NOT true that TB is at least as good as S since S is preferred to TB.. b.
Christine suggests that they first decide on Hals and Subway and then put the winner up against
Taco Bell. Is this a good suggestion? For whom?
[ANSWER] This is a great suggestion for Christine since they will get that H wins over S and then when we put up H against TB they get TB as the winner! She gets her best alternative. Gerrit,
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however, is none too pleased. c.
Greg argues that he should get to set the voting order. Why would he argue for that?
[ANSWER] The voting order matters for the family’s ranking since the family’s ranking is not transitive. If Greg sets the voting order from the pair, (Hals Deli, Subway), his best, Subway, will be decided. d.
Gerrit decides that if Christine is going to set the voting order, since she is the oldest and the oldest always gets to set the order, then he is going to cheat. How might Gerrit cheat and if his assumption that Greg and Christine won't cheat is correct, can be benefit from cheating?
[ANSWER] Suppose the first pair is Hals Deli and Subway. Gerrit’s worst is Taco Bell. If he votes to Hals Deli on the first pair, the family decides Taco Bell at the end. However, the family decides
Subway if Gerrit votes to Subway instead of Hals Deli. Thus, he gets benefits from cheating.
8.
Professor Goodheart always gives two midterms in his communications class. He only uses the higher of the two scores that a student gets on the midterms when he calculates the course grade. a.
Nancy Lerner wants to maximize her grade in this course. Let x be her score on the first midterm and let y be her score on the second midterm. Which combination of scores would Nancy prefer, (x
= 20 and y = 70) or (x = 60 and y = 60)?
[ANSWER] Nancy gets 70 with the combination of scores, (x = 20 and y = 70) and 60 with (x = 60 and y = 60). The score, 70, gives her the higher grade. Thus, she prefers (x = 20 and y = 70). b.
On a graph draw an indifference curve showing all of the combinations of scores that Nancy likes exactly as much as (x = 20 and y = 70). Also draw an indifference curve showing the combinations that Nancy likes exactly as much as (x = 60 and y = 60).
[ANSWER]
Nancy's Indifference Curves y
80
70
60
50
40
30
20
10
0
(20,70)
(60,60)
IC0 IC1
0 10 20 30 40 50 60 70 80 x c.
Does Nancy have convex preferences over these combinations?
[ANSWER] No, Nancy doesn’t have convex preferences. Her indifference curves bow out away
6

from the origin. Suppose a convex combination of (20, 70) and (70, 20), (θ20+(1- θ)70, θ70+(1-
θ)20). Let θ=0.5. Then the convex combination is (45, 45) which gives her 45 scores for grading.
If she has convex preferences, she prefers (45, 45) to (20, 70) and (70, 20). But she would prefer 70 scores from (20, 70) to 45 scores from (45, 45). d.
Nancy is also taking a course in economics form Professor Stern. Professor Stern gives two midterms. Instead of discarding the lower grade, Professor Stern discards the higher one. Let x = score on midterm 1 and let y = score on midterm 2. Which combination of scores would Nancy prefer, (x = 20 and y = 70) or (x = 60 and y = 50).
[ANSWER] Nancy gets 20 score for grading from (20, 70) and 50 from (60, 50). She would prefer
(60, 50) to (20, 70). e.
On a graph, draw an indifference curve showing all of the combinations of scores on her econ exams that Nancy likes exactly as well as (x = 20 and y = 70). Also draw an indifference curve showing the combinations that Nancy likes exactly as well as (x = 60 and y = 50).
[ANSWER]
Nancy's Indifference Curves y
80
70
60
50
40
30
20
10
0
(20,70)
(60,50)
0 10 20 30 40 50 60 70 80 90 100 x f.
Does Nancy have convex preference over these combinations?
[ANSWER] Yes, Nancy’s ICs don’t bow-out away from origin. g.
Can you suggest a utility function to use for Nancy’s communications class? And one for her economics class?
[ANSWER]
A utility function for her communication class: u = max {x, y}
A utility function economics class: u = min {x, y}
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