Unlock document.
This document is partially blurred.
Unlock all pages and 1 million more documents.
Get Access
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
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
yyyyyyyyyyyyyyyyyyyyy
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
