978-0393123982 Chapter 7 Solution Manual Part 1

Chapter 7 NAME

Revealed Preference

Introduction. In the last section, you were given a consumer’s pref-

erences and then you solved for his or her demand behavior. In this

chapter we turn this process around: you are given information about a

consumer’s demand behavior and you must deduce something about the

consumer’s preferences. The main tool is the weak axiom of revealed pref-

erence. This axiom says the following. If a consumer chooses commodity

bundle Awhen she can afford bundle B, then she will never choose bundle

Bfrom any budget in which she can also afford A. The idea behind this

axiom is that if you choose Awhen you could have had B, you must like

Abetter than B. But if you like Abetter than B, then you will never

choose Bwhen you can have A. If somebody chooses Awhen she can

afford B,wesaythatforher,Ais directly revealed preferred to B.The

weak axiom says that if Ais directly revealed preferred to B, then Bis

not directly revealed preferred to A.

Example: Let us look at an example of how you check whether one bundle

is revealed preferred to another. Suppose that a consumer buys the bundle

(xA

1,x

A

2)=(2,3) at prices (pA

1,p

A

2)=(1,4). The cost of bundle (xA

1,x

A

2)

at these prices is (2 ×1) + (3 ×4) = 14. Bundle (2,3) is directly revealed

preferred to all the other bundles that she can afford at prices (1,4), when

she has an income of 14. For example, the bundle (5,2) costs only 13 at

prices (1,4), so we can say that for this consumer (2,3) is directly revealed

preferred to (1,4).

You will also have some problems about price and quantity indexes.

A price index is a comparison of average price levels between two different

times or two different places. If there is more than one commodity, it is not

necessarily the case that all prices changed in the same proportion. Let us

suppose that we want to compare the price level in the “current year” with

the price level in some “base year.” One way to make this comparison

is to compare the costs in the two years of some “reference” commodity

bundle. Two reasonable choices for the reference bundle come to mind.

One possibility is to use the current year’s consumption bundle for the

reference bundle. The other possibility is to use the bundle consumed

in the base year. Typically these will be different bundles. If the base-

year bundle is the reference bundle, the resulting price index is called the

Laspeyres price index. If the current year’s consumption bundle is the

reference bundle, then the index is called the Paasche price index.

Example: Suppose that there are just two goods. In 1980, the prices

were (1,3) and a consumer consumed the bundle (4,2). In 1990, the

prices were (2,4) and the consumer consumed the bundle (3,3). The cost

of the 1980 bundle at 1980 prices is (1 ×4) + (3 ×2) = 10.The cost of this

same bundle at 1990 prices is (2 ×4) + (4 ×2) = 16. If 1980 is treated

as the base year and 1990 as the current year, the Laspeyres price ratio

82 REVEALED PREFERENCE (Ch. 7)

is 16/10. To calculate the Paasche price ratio, you find the ratio of the

cost of the 1990 bundle at 1990 prices to the cost of the same bundle at

1980 prices. The 1990 bundle costs (2 ×3) + (4 ×3) = 18 at 1990 prices.

The same bundle cost (1 ×3) + (3 ×3) = 12 at 1980 prices. Therefore

thePaaschepriceindexis18/12. Notice that both price indexes indicate

that prices rose, but because the price changes are weighted differently,

the two approaches give different price ratios.

Making an index of the “quantity” of stuff consumed in the two

periods presents a similar problem. How do you weight changes in the

amount of good 1 relative to changes in the amount of good 2? This time

we could compare the cost of the two periods’ bundles evaluated at some

Example: In the example above, the Laspeyres quantity index is the ratio

of the cost of the 1990 bundle at 1980 prices to the cost of the 1980 bundle

at 1980 prices. The cost of the 1990 bundle at 1980 prices is 12 and the

cost of the 1980 bundle at 1980 prices is 10, so the Laspeyres quantity

index is 12/10. The cost of the 1990 bundle at 1990 prices is 18 and

the cost of the 1980 bundle at 1990 prices is 16. Therefore the Paasche

quantity index is 18/16.

When you have completed this section, we hope that you will be able

to do the following:

•Given price and consumption data, calculate Paasche and Laspeyres

price and quantity indexes.

•Use the idea of revealed preference to make comparisons of well-being

across time and across countries.

7.1 (0) When prices are (4,6), Goldie chooses the bundle (6,6), and

when prices are (6,3), she chooses the bundle (10,0).

(a) On the graph below, show Goldie’s first budget line in red ink and

(b) Is Goldie’s behavior consistent with the weak axiom of revealed pref-

NAME 83

0 5 10 15 20

5

10

15

Good 1

Good 2

20

a

b

Blue line

Red line

7.2 (0) Freddy Frolic consumes only asparagus and tomatoes, which are

highly seasonal crops in Freddy’s part of the world. He sells umbrellas for

a living, which provides a ﬂuctuating income depending on the weather.

But Freddy doesn’t mind; he never thinks of tomorrow, so each week he

spends as much as he earns. One week, when the prices of asparagus and

tomatoes were each $1 a pound, Freddy consumed 15 pounds of each. Use

blue ink to show the budget line in the diagram below. Label Freddy’s

consumption bundle with the letter A.

(b) The next week the price of tomatoes rose to $2 a pound, but the price

of asparagus remained at $1 a pound. By chance, Freddy’s income had

changed so that his old consumption bundle of (15,15) was just affordable

at the new prices. Use red ink to draw this new budget line on the graph

(c) How much asparagus can he afford now if he spent all of his income

84 REVEALED PREFERENCE (Ch. 7)

(e) Use pencil to shade the bundles of goods on Freddy’s new red budget

line that he definitely will not purchase with this budget. Is it possible

that he would increase his consumption of tomatoes when his budget

0102030

40

10

20

30

Asparagus

Tomatoes

40



a

Blue line

Red line

Pencil

shading

7.3 (0) Pierre consumes bread and wine. For Pierre, the price of bread

is 4 francs per loaf, and the price of wine is 4 francs per glass. Pierre has

an income of 40 francs per day. Pierre consumes 6 glasses of wine and 4

loaves of bread per day.

(a) If Bob and Pierre have the same tastes, can you tell whether Bob is

better off than Pierre or vice versa? Explain. Bob is better

(b) Suppose prices and incomes for Pierre and Bob are as above and that

Pierre’s consumption is as before. Suppose that Bob spends all of his

income. Give an example of a consumption bundle of wine and bread such

that, if Bob bought this bundle, we would know that Bob’s tastes are not

the same as Pierre’s tastes. 7.5 wine and 0 bread, for

example. If they had the same preferences,

NAME 85

7.4 (0) Here is a table of prices and the demands of a consumer named

Ronald whose behavior was observed in 5 different price-income situa-

tions.

Situation p1p2x1x2

A 1 1 5 35

B 1 2 35 10

C 1 1 10 15

D 3 1 5 15

E 1 2 10 10

(a) Sketch each of his budget lines and label the point chosen in each case

(b) Is Ronald’s behavior consistent with the Weak Axiom of Revealed

(c) Shade lightly in red ink all of the points that you are certain are worse

(d) Suppose that you are told that Ronald has convex and monotonic

preferences and that he obeys the strong axiom of revealed preference.

Shade lightly in blue ink all of the points that you are certain are at least

as good as the bundle C.

0102030

40

10

20

30

x1

x2

40



yyyyyyyyyyyyyyyyyyyyy













e

dc

b

a

Red shading

Blue shading

86 REVEALED PREFERENCE (Ch. 7)

7.5 (0) Horst and Nigel live in different countries. Possibly they have

different preferences, and certainly they face different prices. They each

consume only two goods, xand y. Horst has to pay 14 marks per unit of

xand 5 marks per unit of y. Horst spends his entire income of 167 marks

on 8 units of xand 11 units of y. Good xcosts Nigel 9 quid per unit and

good ycosts him 7 quid per unit. Nigel buys 10 units of xand 9 units of

y.

(a) Which prices and income would Horst prefer, Nigel’s income and prices

or his own, or is there too little information to tell? Explain your answer.

(b) Would Nigel prefer to have Horst’s income and prices or his own, or

7.6 (0) Here is a table that illustrates some observed prices and choices

for three different goods at three different prices in three different situa-

tions.

Situation p1p2p3x1x2x3

A 1 2 8 2 1 3

B 4 1 8 3 4 2

C 3 1 2 2 6 2

(a) We will fill in the table below as follows. Where iand jstand for any

of the letters A, B, and C in Row iand Column jof the matrix, write

the value of the Situation-jbundle at the Situation-iprices. For example,

in Row A and Column A, we put the value of the bundle purchased in

Situation A at Situation A prices. From the table above, we see that in

Situation A, the consumer bought bundle (2,1,3) at prices (1,2,8). The

cost of this bundle A at prices A is therefore (1×2)+(2×1)+(8×3) = 28,

so we put 28 in Row A, Column A. In Situation B the consumer bought

bundle (3,4,2). The value of the Situation-B bundle, evaluated at the

situation-A prices is (1 ×3) + (2 ×4) + (8 ×2) = 27, so put 27 in Row

A, Column B. We have filled in some of the boxes, but we leave a few for

you to do.

NAME 87

Prices/Quantities A B C

A28 27 30

B33 32 30

C13 17 16

(b) Fill in the entry in Row iand Column jof the table below with a Dif

the Situation-ibundle is directly revealed preferred to the Situation-jbun-

dle. For example, in Situation A the consumer’s expenditure is $28. We

see that at Situation-A prices, he could also afford the Situation-B bun-

dle, which cost 27. Therefore the Situation-A bundle is directly revealed

preferred to the Situation-B bundle, so we put a Din Row A, Column

B. Now let us consider Row B, Column A. The cost of the Situation-B

bundle at Situation-B prices is 32. The cost of the Situation-A bundle

at Situation-B prices is 33. So, in Situation B, the consumer could not

afford the Situation-A bundle. Therefore Situation B is not directly re-

vealed preferred to Situation A. So we leave the entry in Row B, Column

A blank. Generally, there is a Din Row iColumn jif the number in the

ij entry of the table in part (a) is less than or equal to the entry in Row

i, Column i. There will be a violation of WARP if for some iand j,there

is a Din Row iColumn jand also a Din Row j, Column i.Dothese

observations violate WARP? No.

Situation A B C

A — D I

BI— D

CD I —

(c) Now fill in Row i, Column jwith an Iif observation iis indirectly

revealed preferred to j. Do these observations violate the Strong Axiom

7.7 (0) It is January, and Joe Grad, whom we met in Chapter 5, is

shivering in his apartment when the phone rings. It is Mandy Manana,

one of the students whose price theory problems he graded last term.

Mandy asks if Joe would be interested in spending the month of February

in her apartment. Mandy, who has switched majors from economics to

political science, plans to go to Aspen for the month and so her apartment

will be empty (alas). All Mandy asks is that Joe pay the monthly service

charge of $40 charged by her landlord and the heating bill for the month

of February. Since her apartment is much better insulated than Joe’s,

it only costs $1 per month to raise the temperature by 1 degree. Joe

88 REVEALED PREFERENCE (Ch. 7)

thanks her and says he will let her know tomorrow. Joe puts his earmuffs

back on and muses. If he accepts Mandy’s offer, he will still have to pay

rent on his current apartment but he won’t have to heat it. If he moved,

heating would be cheaper, but he would have the $40 service charge. The

outdoor temperature averages 20 degrees Fahrenheit in February, and it

costs him $2 per month to raise his apartment temperature by 1 degree.

Joe is still grading homework and has $100 a month left to spend on food

and utilities after he has paid the rent on his apartment. The price of

food is still $1 per unit.

(a) Draw Joe’s budget line for February if he moves to Mandy’s apartment

(b) After drawing these lines himself, Joe decides that he would be better

off not moving. From this, we can tell, using the principle of revealed

preference that Joe must plan to keep his apartment at a temperature of

(c) Joe calls Mandy and tells her his decision. Mandy offers to pay half

the service charge. Draw Joe’s budget line if he accepts Mandy’s new

offer. Joe now accepts Mandy’s offer. From the fact that Joe accepted

this offer we can tell that he plans to keep the temperature in Mandy’s

0 10 20 30 40 50 60 70 80

20

40

60

80

100

120

Food

Don’t move budget line

Move budget line

‘New offer’

budget line

Temperature

7.8 (0) Lord Peter Pommy is a distinguished criminologist, schooled

in the latest techniques of forensic revealed preference. Lord Peter is in-

vestigating the disappearance of Sir Cedric Pinchbottom who abandoned

his aging mother on a street corner in Liverpool and has not been seen