Revealed Preference Analysis

1
Revealed Preference Analysis

• Suppose we observe the choices that a consumer
  makes for different budgets (p1, x1),, (pT,
  xT)
• Are the obs. consistent with constrained utility
  maximisation?
• Can we retrieve the utility function that might
  have generated the obs?

2
Assumptions on Preferences

• Preferences
• strictly convex.
• monotonic.
• Together, convexity and monotonicity imply that
  the most preferred affordable bundle is unique.

3
Assumptions on Preferences
x2
If preferences are convex andmonotonic (i.e.
well-behaved)then the most preferredaffordable
bundle is unique.
x2
x1
x1
4
Direct Preference Revelation

• Suppose that given prices p - the bundle x is
  chosen and the bundle y is also affordable, i.e.
• py lt px
• Then x is revealed directly as preferred to y.

5
Direct Preference Revelation
x2
The chosen bundle x isrevealed directly as
preferredto the bundles y and z.
x
z
y
x1
6
Direct Preference Revelation

• Summarising if x is chosen at prices p and y
  is affordable at prices p (i.e. py lt px), then
  prices and choices directly reveal that x is
  preferred to y.
• We write this as x RD y

7
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
  preferred to z. Write this as
  x R zso x R y and y R z x
  R z.

8
Indirect Preference Revelation
z is not affordable when x is chosen.x is not
affordable when y is chosen.
x2
But x RD yand y RD z so x
R z.
x
z
y
x1
9
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 RD y not (y RD
  x).

10
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.

11
The Weak Axiom of Revealed Preference (WARP)
x2
x is chosen when y is availableso x RD y.
y is chosen when x is availableso y RD x.
These obs. violate WARP
y
x
x1
12
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?

13
Checking if Data Violate the WARP
Circles surround affordable bundles thatwere not
chosen.
14
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 R y or x RD y not(y R x or y RD x)

15
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)

16
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).
17
The Strong Axiom of Revealed Preference
The data do not violate the WARP.
18
The Strong Axiom of Revealed Preference
We have thatA RD C, B RD A and C RD Bso, by
transitivity,A R B, B R C and C R A.
I
I
I
The data do not violate the WARP but ...
19
The Strong Axiom of Revealed Preference
The data do not violatethe WARP but there are3
violations of the SARP.
I
I
I
20
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 (Afriat Th.)
• So our 3 data cannot be rationalized by a
  well-behaved preference relation.

21
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)?

22
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

23
Recovering Indifference Curves
Both direct and indirect revelations
neitherWARP nor SARP are violated by the data.
24
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
25
Recovering Indifference Curves
x2
A (p1,p2)(1,1) (x1,x2)(15,15).
A is directly revealed preferredto any bundle in
A
x1
26
Recovering Indifference Curves
x2
A is directly revealed preferred to B and
B
A
x1
27
Recovering Indifference Curves
x2
B is directly revealed preferredto all bundles in
B
x1
28
Recovering Indifference Curves
x2
so, by transitivity, A is indirectlyrevealed
preferred to all bundles in
B
x1
29
Recovering Indifference Curves
x2
so A is now revealed preferredto all bundles in
the union.
B
A
x1
30
Recovering Indifference Curves
x2
Do the same trick with bundle C
Therefore the indifferencecurve containing A
must lie everywhere else above
this shaded set.
B
A
C
x1
31
Recovering Indifference Curves
x2
D is directly revealed preferredto A.
A
D
x1
32
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
33
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
34
Recovering Indifference Curves
x2
bundles revealed to be strictly preferred to A
A
D
x1
35
Recovering Indifference Curves
x2
E is directly revealed preferredto A.
E
A
x1
36
Recovering Indifference Curves
x2
All bundles revealedto be preferred to A
E
B
A
C
D
x1
37
Recovering Indifference Curves
x2
All bundles revealedto be preferred to A
A
x1
All bundles revealed to be less preferred to A
38
Recovering Indifference Curves
x2
The region in which the indifference curve
containing bundle A must lie.
A
x1