Revealed Preference

1
7

• Revealed Preference

2
Revealed Preference Analysis

• Suppose we observe the demands (consumption
  choices) that a consumer makes for different
  budgets. This reveals information about the
  consumers preferences. We can use this
  information to ...

3
Revealed Preference Analysis

• Test the behavioral hypothesis that a consumer
  chooses the most preferred bundle from those
  available.
• Discover the consumers preference relation.

4
Assumptions on Preferences

• Preferences
• do not change while the choice data are gathered.
• are strictly convex.
• are monotonic.
• Together, convexity and monotonicity imply that
  the most preferred affordable bundle is unique.

5
Assumptions on Preferences
x2
If preferences are convex andmonotonic (i.e.
well-behaved)then the most preferredaffordable
bundle is unique.
x2
x1
x1
6
Direct Preference Revelation

• Suppose that the bundle x is chosen when the
  bundle y is affordable. Then x is revealed
  directly as preferred to y (otherwise y would
  have been chosen).

7
Direct Preference Revelation
x2
The chosen bundle x isrevealed directly as
preferredto the bundles y and z.
x
z
y
x1
8
Direct Preference Revelation

• That x is revealed directly as preferred to y
  will be written as
  x y.

9
Indirect Preference Revelation

• Suppose x is revealed directly preferred to y,
  and y is revealed directly preferred to z. Then,
  by transitivity, x is revealed indirectly as
  preferred to z. Write this as
  x zso x y and y z
  x z.

p
I
p
I
10
Indirect Preference Revelation
x2
z is not affordable when x is chosen.
x
z
x1
11
Indirect Preference Revelation
x2
x is not affordable when y is chosen.
x
y
z
x1
12
Indirect Preference Revelation
x2
z is not affordable when x is chosen.x is not
affordable when y is chosen.
x
y
z
x1
13
Indirect Preference Revelation
x2
z is not affordable when x is chosen.x is not
affordable when y is chosen. So x and
z cannot be compared directly.
x
y
z
x1
14
Indirect Preference Revelation
x2
z is not affordable when x is chosen.x is not
affordable when y is chosen. So x and
z cannot be compared directly.
x
But xx y
y
z
x1
15
Indirect Preference Revelation
x2
z is not affordable when x is chosen.x is not
affordable when y is chosen. So x and
z cannot be compared directly.
x
But xx yand y z
y
z
x1
16
Indirect Preference Revelation
x2
z is not affordable when x is chosen.x is not
affordable when y is chosen. So x and
z cannot be compared directly.
x
But xx yand y z
so x z.
y
z
p
x1
I
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Two Axioms of Revealed Preference

• To apply revealed preference analysis, choices
  must satisfy two criteria -- the Weak and the
  Strong Axioms of Revealed Preference.

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The Weak Axiom of Revealed Preference (WARP)

• If the bundle x is revealed directly as preferred
  to the bundle y then it is never the case that y
  is revealed directly as preferred to x i.e.
  x y not (y x).

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The Weak Axiom of Revealed Preference (WARP)

• Choice data which violate the WARP are
  inconsistent with economic rationality.
• The WARP is a necessary condition for applying
  economic rationality to explain observed choices.

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The Weak Axiom of Revealed Preference (WARP)

• What choice data violate the WARP?

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The Weak Axiom of Revealed Preference (WARP)
x2
y
x
x1
22
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x y.
y
x
x1
23
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x y.
y is chosen when x is availableso y x.
y
x
x1
24
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x y.
y is chosen when x is availableso y x.
These statements are inconsistent with
each other.
y
x
x1
25
Checking if Data Violate the WARP

• A consumer makes the following choices
• At prices (p1,p2)(2,2) the choice was (x1,x2)
  (10,1).
• At (p1,p2)(2,1) the choice was (x1,x2)
  (5,5).
• At (p1,p2)(1,2) the choice was (x1,x2)
  (5,4).
• Is the WARP violated by these data?

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Checking if Data Violate the WARP
27
Checking if Data Violate the WARP
Red numbers are costs of chosen bundles.
28
Checking if Data Violate the WARP
Circles surround affordable bundles thatwere not
chosen.
29
Checking if Data Violate the WARP
Circles surround affordable bundles thatwere not
chosen.
30
Checking if Data Violate the WARP
Circles surround affordable bundles thatwere not
chosen.
31
Checking if Data Violate the WARP
32
Checking if Data Violate the WARP
33
Checking if Data Violate the WARP
(10,1) is directlyrevealed preferredto (5,4),
but (5,4) isdirectly revealedpreferred to
(10,1),so the WARP isviolated by the data.
34
Checking if Data Violate the WARP
x2
(5,4) (10,1)
(10,1) (5,4)
x1
35
The Strong Axiom of Revealed Preference (SARP)

• If the bundle x is revealed (directly or
  indirectly) as preferred to the bundle y and x ¹
  y, then it is never the case that the y is
  revealed (directly or indirectly) as preferred to
  x i.e. x y or x y

not ( y x or y x ).
p
I
36
The Strong Axiom of Revealed Preference

• What choice data would satisfy the WARP but
  violate the SARP?

37
The Strong Axiom of Revealed Preference

• Consider the following dataA (p1,p2,p3)
  (1,3,10) (x1,x2,x3) (3,1,4)B (p1,p2,p3)
  (4,3,6) (x1,x2,x3) (2,5,3)C (p1,p2,p3)
  (1,1,5) (x1,x2,x3) (4,4,3)

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The Strong Axiom of Revealed Preference
A (1,3,10) (3,1,4).
B (4,3,6) (2,5,3).
C (1,1,5) (4,4,3).
39
The Strong Axiom of Revealed Preference
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The Strong Axiom of Revealed Preference
In situation A,bundle A is directly
revealedpreferred tobundle C A C.
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The Strong Axiom of Revealed Preference
In situation B,bundle B is directly
revealedpreferred tobundle A B A.
42
The Strong Axiom of Revealed Preference
In situation C,bundle C is directly
revealedpreferred tobundle B C B.
43
The Strong Axiom of Revealed Preference
44
The Strong Axiom of Revealed Preference
The data do not violate the WARP.
45
The Strong Axiom of Revealed Preference
We have thatA C, B A and C Bso,
by transitivity,A B, B C and C A.
The data do not violate the WARP but ...
46
The Strong Axiom of Revealed Preference
We have thatA C, B A and C Bso,
by transitivity,A B, B C and C A.
I
I
I
The data do not violate the WARP but ...
47
The Strong Axiom of Revealed Preference
B A is inconsistentwith A B.
I
I
I
The data do not violate the WARP but ...
48
The Strong Axiom of Revealed Preference
A C is inconsistentwith C A.
I
I
I
The data do not violate the WARP but ...
49
The Strong Axiom of Revealed Preference
C B is inconsistentwith B C.
I
I
I
The data do not violate the WARP but ...
50
The Strong Axiom of Revealed Preference
The data do not violatethe WARP but there are3
violations of the SARP.
I
I
I
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The Strong Axiom of Revealed Preference

• That the observed choice data satisfy the SARP is
  a condition necessary and sufficient for there to
  be a well-behaved preference relation that
  rationalizes the data.
• So our 3 data cannot be rationalized by a
  well-behaved preference relation.

52
Recovering Indifference Curves

• Suppose we have the choice data satisfy the SARP.
• Then we can discover approximately where are the
  consumers indifference curves.
• How?

53
Recovering Indifference Curves

• Suppose we observeA (p1,p2) (1,1)
  (x1,x2) (15,15)B (p1,p2) (2,1) (x1,x2)
  (10,20)C (p1,p2) (1,2) (x1,x2)
  (20,10)D (p1,p2) (2,5) (x1,x2)
  (30,12)E (p1,p2) (5,2) (x1,x2) (12,30).
• Where lies the indifference curve containing the
  bundle A (15,15)?

54
Recovering Indifference Curves

• The table showing direct preference revelations
  is

55
Recovering Indifference Curves
Direct revelations only the WARPis not violated
by the data.
56
Recovering Indifference Curves

• Indirect preference revelations add no extra
  information, so the table showing both direct and
  indirect preference revelations is the same as
  the table showing only the direct preference
  revelations

57
Recovering Indifference Curves
Both direct and indirect revelations
neitherWARP nor SARP are violated by the data.
58
Recovering Indifference Curves

• Since the choices satisfy the SARP, there is a
  well-behaved preference relation that
  rationalizes the choices.

59
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20)C (p1,p2)(1,2)
(x1,x2)(20,10)D (p1,p2)(2,5)
(x1,x2)(30,12)E (p1,p2)(5,2) (x1,x2)(12,30).
E
B
D
A
C
x1
60
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20)C (p1,p2)(1,2)
(x1,x2)(20,10)D (p1,p2)(2,5)
(x1,x2)(30,12)E (p1,p2)(5,2) (x1,x2)(12,30).
E
B
D
A
C
x1
Begin with bundles revealedto be less preferred
than bundle A.
61
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15).
A
x1
62
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15).
A
x1
63
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15).
A is directly revealed preferredto any bundle in
A
x1
64
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20).
E
B
D
A
C
x1
65
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20).
B
A
x1
66
Recovering Indifference Curves
x2
A is directly revealed preferred to B and
B
A
x1
67
Recovering Indifference Curves
x2
B is directly revealed preferredto all bundles in
B
x1
68
Recovering Indifference Curves
x2
so, by transitivity, A is indirectlyrevealed
preferred to all bundles in
B
x1
69
Recovering Indifference Curves
x2
so A is now revealed preferredto all bundles in
the union.
B
A
x1
70
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)C
(p1,p2)(1,2) (x1,x2)(20,10).
E
B
D
A
C
x1
71
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)C
(p1,p2)(1,2) (x1,x2)(20,10).
A
C
x1
72
Recovering Indifference Curves
x2
A is directly revealedpreferred to C and ...
A
C
x1
73
Recovering Indifference Curves
x2
C is directly revealed preferredto all bundles in
C
x1
74
Recovering Indifference Curves
x2
so, by transitivity, A isindirectly revealed
preferredto all bundles in
C
x1
75
Recovering Indifference Curves
x2
so A is now revealed preferredto all bundles in
the union.
B
A
C
x1
76
Recovering Indifference Curves
x2
so A is now revealed preferredto all bundles in
the union.
Therefore the indifferencecurve containing A
must lie everywhere else above
this shaded set.
B
A
C
x1
77
Recovering Indifference Curves

• Now, what about the bundles revealed as more
  preferred than A?

78
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20)C (p1,p2)(1,2)
(x1,x2)(20,10)D (p1,p2)(2,5)
(x1,x2)(30,12)E (p1,p2)(5,2) (x1,x2)(12,30).
E
B
A
D
C
A
x1
79
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)D
(p1,p2)(2,5) (x1,x2)(30,12).
A
D
x1
80
Recovering Indifference Curves
x2
D is directly revealed preferredto A.
A
D
x1
81
Recovering Indifference Curves
x2
D is directly revealed preferredto
A.Well-behaved preferences areconvex
A
D
x1
82
Recovering Indifference Curves
x2
D is directly revealed preferredto
A.Well-behaved preferences areconvex so all
bundles on the line between A and D are
preferred to A also.
A
D
x1
83
Recovering Indifference Curves
x2
D is directly revealed preferredto
A.Well-behaved preferences areconvex so all
bundles on the line between A and D are
preferred to A also.
A
D
As well, ...
x1
84
Recovering Indifference Curves
x2
all bundles containing thesame amount of
commodity 2and more of commodity 1 thanD are
preferred to D and therefore are preferred
to A also.
A
D
x1
85
Recovering Indifference Curves
x2
bundles revealed to be strictly preferred to A
A
D
x1
86
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)B
(p1,p2)(2,1) (x1,x2)(10,20)C (p1,p2)(1,2)
(x1,x2)(20,10)D (p1,p2)(2,5)
(x1,x2)(30,12)E (p1,p2)(5,2) (x1,x2)(12,30).
E
B
A
D
C
A
x1
87
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15)E
(p1,p2)(5,2) (x1,x2)(12,30).
E
A
x1
88
Recovering Indifference Curves
x2
E is directly revealed preferredto A.
E
A
x1
89
Recovering Indifference Curves
x2
E is directly revealed preferredto
A.Well-behaved preferences areconvex
E
A
x1
90
Recovering Indifference Curves
x2
E is directly revealed preferredto
A.Well-behaved preferences areconvex so all
bundles on the line between A and E are
preferred to A also.
E
A
x1
91
Recovering Indifference Curves
x2
E is directly revealed preferredto
A.Well-behaved preferences areconvex so all
bundles on the line between A and E are
preferred to A also.
E
A
As well, ...
x1
92
Recovering Indifference Curves
x2
all bundles containing thesame amount of
commodity 1and more of commodity 2 thanE are
preferred to E and therefore are preferred
to A also.
E
A
x1
93
Recovering Indifference Curves
x2
More bundles revealed to be strictly preferred to
A
E
A
x1
94
Recovering Indifference Curves
x2
Bundles revealedearlier as preferredto A
E
B
A
C
D
x1
95
Recovering Indifference Curves
x2
All bundles revealedto be preferred to A
E
B
A
C
D
x1
96
Recovering Indifference Curves

• Now we have upper and lower bounds on where the
  indifference curve containing bundle A may lie.

97
Recovering Indifference Curves
x2
All bundles revealedto be preferred to A
A
x1
All bundles revealed to be less preferred to A
98
Recovering Indifference Curves
x2
All bundles revealedto be preferred to A
A
x1
All bundles revealed to be less preferred to A
99
Recovering Indifference Curves
x2
The region in which the indifference curve
containing bundle A must lie.
A
x1
100
Index Numbers

• Over time, many prices change. Are consumers
  better or worse off overall as a consequence?
• Index numbers give approximate answers to such
  questions.

101
Index Numbers

• Two basic types of indices
• price indices, and
• quantity indices
• Each index compares expenditures in a base period
  and in a current period by taking the ratio of
  expenditures.

102
Quantity Index Numbers

• A quantity index is a price-weighted average of
  quantities demanded i.e.
• (p1,p2) can be base period prices (p1b,p2b) or
  current period prices (p1t,p2t).

103
Quantity Index Numbers

• If (p1,p2) (p1b,p2b) then we have the Laspeyres
  quantity index

104
Quantity Index Numbers

• If (p1,p2) (p1t,p2t) then we have the Paasche
  quantity index

105
Quantity Index Numbers

• How can quantity indices be used to make
  statements about changes in welfare?

106
Quantity Index Numbers

• If
  thenso consumers overall were better off in
  the base period than they are now in the current
  period.

107
Quantity Index Numbers

• If
  thenso consumers overall are better off in
  the current period than in the base period.

108
Price Index Numbers

• A price index is a quantity-weighted average of
  prices i.e.
• (x1,x2) can be the base period bundle (x1b,x2b)
  or else the current period bundle (x1t,x2t).

109
Price Index Numbers

• If (x1,x2) (x1b,x2b) then we have the Laspeyres
  price index

110
Price Index Numbers

• If (x1,x2) (x1t,x2t) then we have the Paasche
  price index

111
Price Index Numbers

• How can price indices be used to make statements
  about changes in welfare?
• Define the expenditure ratio

112
Price Index Numbers

• Ifthenso consumers overall are better off
  in the current period.

113
Price Index Numbers

• But, ifthenso consumers overall were
  better off in the base period.

114
Full Indexation?

• Changes in price indices are sometimes used to
  adjust wage rates or transfer payments. This is
  called indexation.
• Full indexation occurs when the wages or
  payments are increased at the same rate as the
  price index being used to measure the aggregate
  inflation rate.

115
Full Indexation?

• Since prices do not all increase at the same
  rate, relative prices change along with the
  general price level.
• A common proposal is to index fully Social
  Security payments, with the intention of
  preserving for the elderly the purchasing power
  of these payments.

116
Full Indexation?

• The usual price index proposed for indexation is
  the Paasche quantity index (the Consumers Price
  Index).
• What will be the consequence?

117
Full Indexation?
Notice that this index uses currentperiod prices
to weight both base andcurrent period
consumptions.
118
Full Indexation?
x2
Base period budget constraint
Base period choice
x2b
x1
x1b
119
Full Indexation?
x2
Base period budget constraint
Base period choice
x2b
Current period budgetconstraint before indexation
x1
x1b
120
Full Indexation?
x2
Base period budget constraint
Base period choice
Current period budgetconstraint after full
indexation
x2b
x1
x1b
121
Full Indexation?
x2
Base period budget constraint
Base period choice
Current period budgetconstraint after indexation
x2b
Current period choiceafter indexation
x1
x1b
122
Full Indexation?
x2
Base period budget constraint
Base period choice
Current period budgetconstraint after indexation
x2b
Current period choiceafter indexation
x2t
x1
x1b
x1t
123
Full Indexation?
x2
(x1t,x2t) is revealed preferred to(x1b,x2b) so
full indexation makesthe recipient strictly
better off if relative prices change betweenthe
base and current periods.
x2b
x2t
x1
x1b
x1t
124
Full Indexation?

• So how large is this bias in the US CPI?
• A table of recent estimates of the bias is given
  in the Journal of Economic Perspectives, Volume
  10, No. 4, p. 160 (1996). Some of this list of
  point and interval estimates are as follows

125
Full Indexation?
126
Full Indexation?

• So suppose a social security recipient gained by
  1 per year for 20 years.
• Q How large would the bias have become at the
  end of the period?

127
Full Indexation?

• So suppose a social security recipient gained by
  1 per year for 20 years.
• Q How large would the bias have become at the
  end of the period?
• A
  so after 20 years social security payments would
  be about 22 too large.