Revealed Preference

1
Revealed Preference

• Molly W. Dahl
• Georgetown University
• Econ 101 Spring 2008

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 DRP 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 IRP zso x DRP y and y DRP z
  x IRP z.

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 x DRP 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 x DRP yand y DRP 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 x DRP yand y DRP z so
x IRP z.
y
z
x1
17
Two Axioms of Revealed Preference

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

18
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 DRP y not (y DRP x).

19
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.

20
The Weak Axiom of Revealed Preference (WARP)

• What choice data violate the WARP?

21
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 DRP y.
y
x
x1
23
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x DRP y.
y is chosen when x is availableso y DRP x.
y
x
x1
24
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x DRP y.
y is chosen when x is availableso y DRP 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?

26
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) DRP (10,1)
(10,1) DRP (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 DRP y or x IRP y

not ( y DRP x or y IRP x ).
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)

38
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
40
The Strong Axiom of Revealed Preference
In situation A,bundle A is directly
revealedpreferred tobundle C A DRP C.
41
The Strong Axiom of Revealed Preference
In situation B,bundle B is directly
revealedpreferred tobundle A B DRP A.
42
The Strong Axiom of Revealed Preference
In situation C,bundle C is directly
revealedpreferred tobundle B C DRP 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 DRP C, B DRP A and C DRP Bso,
by transitivity,A IRP B, B IRP C and C IRP A.
The data do not violate the WARP but ...
46
The Strong Axiom of Revealed Preference
We have thatA DRP C, B DRP A and C DRP Bso,
by transitivity,A IRP B, B IRP C and C IRP A.
I
I
I
The data do not violate the WARP but ...
47
The Strong Axiom of Revealed Preference
B DRP A is inconsistentwith A IRP B.
I
I
I
The data do not violate the WARP but ...
48
The Strong Axiom of Revealed Preference
A DRP C is inconsistentwith C IRP A.
I
I
I
The data do not violate the WARP but ...
49
The Strong Axiom of Revealed Preference
C DRP B is inconsistentwith B IRP 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
51
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
The region in which the indifference curve
containing bundle A must lie.
A
x1