Study Guide 01-02 - Solutions PDF

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Eco 200 Study Guide 01-02 – SolutionsThe questions on the study guide are carefully selected not only for the deepened understanding the solution yields but also for the skills and insights you develop while working out the solution. One skill especially important to learn in Eco200 is how to ``unst...

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Eco 200 Study Guide 01-02 – Solutions The questions on the study guide are carefully selected not only for the deepened understanding the solution yields but also for the skills and insights you develop while working out the solution. One skill especially important to learn in Eco200 is how to ``unstuck" yourself after you have gotten stuck in a problem's solution. To practice this skill, you need to get stuck in the first place. Looking at the solution does not tell you how to figure out the solution, and in most cases bares you from finding out in the future. Therefore, I want to caution you against looking at the solutions too early. Ideally, you want to look at the solutions only after •

You have a solution written out that you are convinced is right and you want to double check your answer.

Short of having an answer of your own, please make sure you have tried at least four of the following approaches before looking at the solution • • • • • • •

• • • •

kfbw

Attempted to solve the problem. Tried to solve the problem with a different approach, e.g., graphically instead of with calculus. Wrote down the definitions of items involved in the problem and explained the problem in English. Tried to find and solve a special case. Made up numbers for all parameters and all but one of the variables, and solve this problem. Made up numbers for all parameters and solved this problem. Changed the problem into a simpler one, e.g., with two consumers instead of three, with a discrete or uniform distribution instead of a complicated continuous one, and tried to solve the simpler problem. Talked to a class mate about the problem. Talked to a peer mentor at the Economic Study Center about the problem. Talked to the course instructor about the problem after tutorials or in office hours. Talked to the professor after class/in office hours.

Solutions to all exercises in the reading are included in the textbook in Appendix B, p. 639 and ff. --------------------------------p. 68 # N1. Every day Fred buys wax lips and candy cigarettes. After deciding how many of each to buy, he multiplies the number of sets of wax lips times the number of packs of candy cigarettes. The higher this number comes out to be, the happier he is. For example, 3 sets of wax lips and 5 packs of candy cigarettes will make him happier than 2 sets of wax lips and 7 packs of candy cigarettes, because 3 x 5 is greater than 2 x 7. Wax lips sell for $2 a pair and candy cigarettes for $1 a pack. Fred has $20 to spend each day. a. Make a table that looks like this: Pairs of Wax Lips Packs of Candy Cigarettes 0 1 2 . . . 10 where each row of the chart corresponds to a basket on Fred’s budget line. Fill in the second column. Answer: The following baskets lie on Fred’s budget line (for every additional wax lip, Fred has to give up two candy cigarettes) Pairs of Wax Lips 0 1 2 3 4 5 6 7 8 9 10

Packs of Candy Cigarettes 20 18 16 14 12 10 8 6 4 2 0

b. Draw a graph showing Fred’s budget line and marking the baskets described by your table. Draw Fred’s indifference curves through these baskets. If he must select among these baskets, which one will Fred choose?

Answer: The following figure shows Fred’s budget line as well as his indifference curves going through each of the baskets listed above. (This figure was generated using Excel by computing several baskets that would generate the same happiness as the respective baskets listed above.)

28 23

Candy Cigarettes

Budget line 18

happiness = 0 happiness = 18

13

happiness = 32 happiness = 42

8

happiness = 48 happiness = 50

3 -2 0

5

10

15

Pairs of Wax Lips

Among those bundles, Fred would chooses to buy 5 pairs of wax lips and 10 packs of candy cigarettes.

c. Add to your table a third column labeled MV for the marginal value of wax lips in terms of candy cigarettes. Fill in the MV for each basket. (Hint: For each basket construct another basket that has one less pair of wax lips but enough more packs of candy cigarettes to be equally desirable. How many packs of candy cigarettes have been added to the basket?) For which basket is the marginal value closest to the relative price of wax lips? Is this consistent with your answer to part b? Answer: The following table computes the marginal value of pairs of wax lips as described in the hint of the question. Fred chooses the basket for which the MV of a pair of wax lips is equal to its relative price, i.e., the price of wax lips expressed in packs of candy cigarettes, which is 2. Thus he takes either the basket (5,10) or the basket (6,8); this is consistent with the answer to b.

Pairs of Wax Lips 0 1 2 3 4 5 6 7 8 9 10

One less Candy Happiness pair of wax Cigarettes lips 20 0 18 18 0 16 32 1 14 42 2 12 48 3 10 50 4 8 48 5 6 42 6 4 32 7 2 18 8 0 0 9

Total candy cigarettes needed to maintain happiness

Additional candy cigarettes needed to maintain happiness

infinite 32 21 16 12.5 9.6 7 4.57 2.25 0

infinite 16 7 4 2.5 1.6 1 0.57 0.25 0

MV

infinite 16 7 4 2.5 1.6 1 0.57 0.25 0

Note: Some of you may have computed a constant marginal value of 2 or -2 by keeping the budget fixed, not the happiness generated by each basket. This essentially computes the slope of the budget line which is, of course, the same for all baskets. Note: There seems to be some ambiguity in the answer to this question. This ambiguity results from the fact that we have only allowed Fred to count candy cigarettes and wax lips in whole number quantities. If we had allowed fractional quantities, we would argue as follows: suppose that Fred now has w pairs of wax lips and p packs of candy cigarettes. Now we reduce his wax lip holdings by h pairs (where h is a very small fraction), and then increase his candy cigarette holdings by m packs, where m is chosen to keep Fred just as happy as he was before. Thus we have: w x p = (w - h) x (p + m). Solving this gives: m ≈ p/w x h. (We ignore the term h x m, which is small.) It follows that the marginal value of wax lips is m = p/w x h packs of candy cigarettes per h pairs of wax lips, or p/w packs of candy cigarettes per pair of wax lips. With this, we can replace the approximations in the table above by the following exact figures:

Pairs of Wax Lips w 0 1 2 3 4 5

Packs of Candy Cigarettes p 20 18 16 14 12 10

MV = p/w x 1 infinite 18 8 4 2/3 3 2

6 7 8 9 10

8 6 4 2 0

1 1/3 6/7 1/2 2/9 0

Now we can see that with 5 pairs of wax lips and 10 packs of candy cigarettes, For Fred, the MV of a pair of wax lips is exactly 2 packs of candy cigarettes, so that this is the basket he will choose.

p. 69 #2 Draw your indifference curves between nickels and dimes, assuming that you are always willing to trade 2 nickels for 1 dime, or vice versa. What is the marginal value of nickels in terms of dimes? Answer: The following table shows how you are willing to trade-off 2 nickel for 1 dimes, along the indifference curve for, say U=50 and U =100, respectively.

Utility 50 50 50 50 50 50

Numbers of nickels 10 8 6 4 2 0

Numbers of dimes 0 1 2 3 4 5

Utility 100 100 100 100 100 100

Number of nickels 20 16 12 8 4 0

Number of dimes 0 2 4 6 8 10

Representing the information given in this table into graphs yields

p. 69 #3 Suppose that you like to own both left and right shoes, but that a right shoe is of no use to you unless you own a matching left one, and vice versa. Draw your indifference curves between left and right shoes. Answer: It is useful to first create a table that shows for each combination of left and right the uses of pairs of shoes (representing utility) we get:

Number of happiness zero left shoe one left shoe two left shoes three left shoes

zero right shoe

one right shoe

two right shoes

three right shoes

0 0 0 0

0 1 1 1

0 1 2 2

0 1 2 3

From that we can read of different combinations that yield the same utility. For example Utility = 1 Numbers Numbers of of right left shoes shoes 1 1 1 2 1 3 2 1 3 1

Utility = 2 Numbers Numbers of right of left shoes shoes 2 2 2 3 2 4 3 2 4 2

Utility = 3 Numbers Numbers of right of left shoes shoes 3 3 3 4 4 3 3 5 3 6

Translating the information given in the table, we find the following indifference curves.

Note: Not all bundles that generate U=3 are “efficient”, indeed the most efficient one is to own 3right shoes and 3left shoes. It is important to separate the elements of analysis (indifference curves, budget lines, …) from the outcome (the optimal consumption bundle).

----------------------------------

Numeric Question 1. In the first class we said that a good first way to approach an unknown problem or to build a model is to make up numbers that capture some essential aspect of the problem or the real-world set-up. The following question asks you to do just that. a. Let’s assume you get 10 oranges, some of which are high quality, some are low quality. Oranges

10 9 high, high, 1 low 0 low quality quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

Happiness o

o

o

Assume that Adam only cares for the number of oranges he consumes, but is indifferent between eating low and high quality oranges. What numbers would represent Adam’s happiness for each of the above combinations? (There are many different answers possible.) Assume that Lydia only cares for the number of high quality oranges she consumes, and derives no utility at all from low quality oranges. What numbers would represent Lydia’s happiness for each of the above combinations? (There are many different answers possible.) Now think about your own happiness consuming each of these different bundles. Assign a happiness or “utility” value to each of these combinations so that the relative values represent your relative enjoyment of each of these bundles.

Answer: Since Adam is indifferent between eating low and high quality oranges and only cares for the number of oranges he consumes all bundles should be assigned the same numeric value – to represent the indifference. Any number would be appropriate. For example: Oranges

Happiness

10 high, 0 low quality 10

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

10

10

10

10

10

10

10

10 high, 0 low quality 1234

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

1234

1234

1234

1234

1234

1234

1234

10

10

10

Or Oranges

Happiness

1234

1234

1234

Or Oranges

Happiness

10 high, 0 low quality 0

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

0

0

0

0

0

0

0

0

0

0

Would all be appropriate answers. What matters is the how the utility changes between bundles and that that pattern is consistent with the information we are given. --------------------------------Next, we are told that Lydia only cares for the number of high quality oranges she consumes, and derives no utility at all from low quality oranges. Thus, her utility should decrease as the number of high quality oranges decreases. We know nothing about how fast the utility declines. So Oranges

Happiness

10 high, 0 low quality 10

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

9

8

6

5

3

1

0

7

4

2

Would be a “natural” way to represent Lydia’s preferences. If you think diminishing marginal utility is reasonable, then Oranges

Happiness

10 high, 0 low quality 560

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

550

530

460

410

270

100

0

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

1 high, 9 low quality

0 high, 10 low quality

500

350

190

Would be appropriate. But even Oranges

10 9 high, high, 1 low 0 low quality quality

8 high, 7 high, 2 low 3 low quality quality

3 high, 2 high, 7 low 8 low quality quality

Happiness

110

105

98

95

80

60

35

32

31

20

18

is valid/ appropriate given the information available. -----------------------------------------Assigning your happiness or utility may result in different number for every individual person. Here are my numbers: Oranges

Happiness

10 high, 0 low quality 95

9 high, 1 low quality

8 high, 7 high, 2 low 3 low quality quality

6 high, 4 low quality

5 high, 4 high, 5 low 6 low quality quality

3 high, 2 high, 7 low 8 low quality quality

1 high, 9 low quality

0 high, 10 low quality

93

87

70

60

34

26

25

79

38

30

What pattern do you see? Your numbers will likely differ. b. Now, more generally people might consume any combination of high and low quality oranges. To represent this, create a table with number of high quality oranges heading the columns and number of low quality oranges describing the rows. Then each cell in the table corresponds to a particular combination of low and high quality oranges. Find numbers that represent your personal happiness for different combinations of high and low quality oranges consumed per month. o What would these numbers look like if you did not care at all for oranges? o What would these numbers look like if you had a serious allergy against oranges? Answer: o

Here are numbers that sort of reflect my happiness from consuming high and low quality oranges. Your numbers, of course, might look differently.

Number of low quality oranges consumed

Number representing happiness 0 1 2 3 4

0 0 8 11 14 16

Number of high Quality oranges consumed 1 2 3 4 16 23 28 32 24 31 36 40 27 34 39 43 30 36 42 46 32 39 44 48

5 36 44 47 50 52

For me personally, I much prefer one orange over none, two oranges over one. But the second orange gives me less additional happiness than the first. This tendency continues with additional oranges. Your preferences might look differently

o

If I didn’t care for oranges at all, then the numbers would all be constant – having more than one orange would not make me more or less happy than having two oranges or none. For example:

Number of low quality oranges consumed

Number representing happiness 0 1 2 3 4

Number of high Quality oranges consumed 1 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0

5 0 0 0 0 0

Note: The utility I derive from having 3 oranges (or really any number of oranges) is the same as the utility I derive from having none. The value chosen does not matter at all – I could have expressed it as constant at 0, -100 or +1000. The key is that the utility does not changes, i.e., in this case it is not true that “more oranges” make me “happier.” o

If I had a severe allergic reaction, I would be worse of the more oranges I consume. Let’s assume that the allergic reaction is as bad after consuming a low quality orange as after consuming a high quality orange. For example:

Number of low quality oranges consumed

Number representing happiness 0 1 2 3 4

0 100 90 75 45 10

Number of high Quality oranges consumed 1 2 3 4 90 75 45 10 75 45 10 -30 45 10 -30 -75 10 -30 -75 -125 -30 -75 -125 -180

5 -30 -75 -125 -180 -240

c. Multiply all numbers you assigned by 100. Does this still represent your happiness (in an economic sense)? Why or why not? Answer: Going back to my own personal happiness as described in the first table, multiplying all numbers by 100 yields

qual ity ora nge

Number representing happiness 0 1

0 0 800

Number of high Quality oranges consumed 1 2 3 1600 2300 2800 2400 3100 3600

4 3200 4000

2 3 4

1100 1400 1600

2700 3000 3200

3400 3600 3900

3900 4200 4400

4300 4600 4800

These numbers also represent my preferences between different bundles of high and low quality oranges, just as the numbers in part b). Expressing “happiness” , i.e., utility, is a way of assigning a number to every possible orange bundle such that more preferred bundles get assigned larger numbers than less-preferred bundles. Multiplying by 100 preserves the order (ranking). Among the bundles shown here, having four high and four low quality oranges is still the highest ranked bundle – just as before. d. Draw indifference (iso-utility) curves that correspond to the numbers you assigned. Answer: Indifference curves plot consumption bundles that yield the same utility (happiness). From table 1 we see, for example, that bundles (3,2) , (1,3), and (0,5) yield the same happiness and hence have to be points on the same indifference line.

Number of high quality oranges consumed

We draw (or use Excel to do it for us ☺)

U = 39 U = 36 U = 16

Number of low quality oranges consumed

e. What are current prices at Kensington market for crappy low quality oranges? For delicious highquality oranges? Draw a few budget (iso-expenditure) lines. Answer: On a rec...