1. Molly W. Dahl Georgetown University Econ 101 – Spring 2009
Revealed Preference
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009

2. Revealed Preference Analysis
Suppose we observe the demands (consumption choices) that a consumer makes for different budgets. This reveals information about the consumer’s preferences. We can use this information to
Test the behavioral hypothesis that a consumer chooses the most preferred bundle from those available.
Discover the consumer’s preference relation.

3. Assumptions on Preferences
do not change while the choice data are gathered.
are strictly convex.
are monotonic.

4. Assumptions on Preferencesx2
If preferences are strictly
convex and monotonic
then the most preferred affordable bundle is unique.
x2*
x1* x1

5. 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).

6. Direct Preference Revelationx2
The chosen bundle x* is revealed directly as preferred to the bundles y and z. x* z y x1

7. Direct Preference Revelation
That x is revealed directly as preferred to y will be written as x DRP y.

8. 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 z so x DRP y and y DRP z x IRP z.

9. Indirect Preference Revelationx2
z is not affordable when x* is chosen. x* z x1

10. Indirect Preference Revelationx2
x* is not affordable when y* is chosen. x* y* z x1

11. Indirect Preference Revelationx2
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

12. Indirect Preference Revelationx2
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* and y* DRP z so x* IRP z. y* z x1

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

14. 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).

15. 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.

16. The Weak Axiom of Revealed Preference (WARP)
What choice data violate the WARP?

17. The Weak Axiom of Revealed Preference (WARP)
x is chosen when y is available so x DRP y. y x x1

18. The Weak Axiom of Revealed Preference (WARP)
x is chosen when y is available so x DRP y.
y is chosen when x is available so y DRP x.
These statements are inconsistent with each other. y x x1

19. 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?

20. Checking if Data Violate the WARP
Red numbers are costs of chosen bundles.

21. Checking if Data Violate the WARP
Circles surround affordable bundles that were not chosen.

22. Checking if Data Violate the WARP

23. Checking if Data Violate the WARP
(10,1) is directly revealed preferred to (5,4), but (5,4) is directly revealed preferred to (10,1), so the WARP is violated by the data.

24. Checking if Data Violate the WARPx2
(5,4) DRP (10,1)
(10,1) DRP (5,4) x1

25. 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 ).

26. The Strong Axiom of Revealed Preference
What choice data would satisfy the WARP but violate the SARP?

27. The Strong Axiom of Revealed Preference
Consider the following data: A: (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)

28. The Strong Axiom of Revealed Preference
B: ($4,$3,$6) (2,5,3).
C: ($1,$1,$5) (4,4,3).

29. The Strong Axiom of Revealed Preference
The data do not violate the WARP.

30. The Strong Axiom of Revealed Preference
We have that A DRP C, B DRP A
and C DRP B so, by transitivity, A IRP B, B IRP C
and C IRP A. I I I
The data do not violate the WARP but ...

31. The Strong Axiom of Revealed Preference
B DRP A is inconsistent with A IRP B. I I I
The data do not violate the WARP but ...

32. The Strong Axiom of Revealed Preference
The data do not violate the WARP but there are 3 violations of the SARP. I I I

33. 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 data cannot be rationalized by a well-behaved preference relation.

34. Recovering Indifference Curves
Suppose we have the choice data satisfy the SARP.
Then we can discover approximately where are the consumer’s indifference curves.
How?

35. Recovering Indifference Curves
Suppose we observe: 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).
Where lies the indifference curve containing the bundle A = (15,15)?

36. Recovering Indifference Curves
The table showing direct preference revelations is:

37. Recovering Indifference Curves
Direct revelations only; the WARP is not violated by the data.

38. Recovering Indifference Curves
In this example, indirect preference revelations add no extra information. The table showing both direct and indirect preference revelations is the same as the table showing only the direct preference revelations:

39. Recovering Indifference Curves
Both direct and indirect revelations; neither WARP nor SARP are violated by the data.

40. Recovering Indifference Curves
Since the choices satisfy the SARP, there is a well-behaved preference relation that “rationalizes” the choices.

41. Recovering Indifference Curvesx2
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 revealed to be less preferred than bundle A.

42. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15). A x1

43. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15).
A is directly revealed preferred to any bundle in A x1

44. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) B: (p1,p2)=(2,1); (x1,x2)=(10,20). E B D A C x1

45. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) B: (p1,p2)=(2,1); (x1,x2)=(10,20). B A x1

46. Recovering Indifference Curvesx2
A is directly revealed preferred
to B and … B A x1

47. Recovering Indifference Curvesx2
B is directly revealed preferred to all bundles in B x1

48. Recovering Indifference Curvesx2
so, by transitivity, A is indirectly revealed preferred to all bundles in B x1

49. Recovering Indifference Curvesx2
so A is now revealed preferred to all bundles in the union. B A x1

50. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) C: (p1,p2)=(1,2); (x1,x2)=(20,10). E B D A C x1

51. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) C: (p1,p2)=(1,2); (x1,x2)=(20,10). A C x1

52. Recovering Indifference Curvesx2
A is directly revealed preferred to C and ... A C x1

53. Recovering Indifference Curvesx2
C is directly revealed preferred to all bundles in C x1

54. Recovering Indifference Curvesx2
so, by transitivity, A is indirectly revealed preferred to all bundles in C x1

55. Recovering Indifference Curvesx2
so A is now revealed preferred to all bundles in the union.
Therefore the indifference curve containing A must lie everywhere else above this shaded set. B A C x1

56. Recovering Indifference Curves
Now, what about the bundles revealed as more preferred than A?

57. Recovering Indifference Curvesx2
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

58. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) D: (p1,p2)=(2,5); (x1,x2)=(30,12). A D x1

59. Recovering Indifference Curvesx2
D is directly revealed preferred to A. A D x1

60. Recovering Indifference Curvesx2
D is directly revealed preferred to A. Well-behaved preferences are convex A D x1

61. Recovering Indifference Curvesx2
D is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and D are preferred to A also. A D x1

62. Recovering Indifference Curvesx2
D is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and D are preferred to A also. A D
As well, ... x1

63. Recovering Indifference Curvesx2
all bundles containing the same amount of commodity 2 and more of commodity 1 than D are preferred to D and therefore are preferred to A also. A D x1

64. Recovering Indifference Curvesx2
bundles revealed to be strictly preferred to A A D x1

65. Recovering Indifference Curvesx2
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

66. Recovering Indifference Curvesx2
A: (p1,p2)=(1,1); (x1,x2)=(15,15) E: (p1,p2)=(5,2); (x1,x2)=(12,30). E A x1

67. Recovering Indifference Curvesx2
E is directly revealed preferred to A. Well-behaved preferences are convex so all bundles on the line between A and E are preferred to A also. E A
As well, ... x1

68. Recovering Indifference Curvesx2
all bundles containing the same amount of commodity 1 and more of commodity 2 than E are preferred to E and therefore are preferred to A also. E A x1

69. Recovering Indifference Curvesx2
More bundles revealed to be strictly preferred to A E A x1

70. Recovering Indifference Curvesx2
Bundles revealed earlier as preferred to A E B A C D x1

71. Recovering Indifference Curvesx2
All bundles revealed to be preferred to A E B A C D x1

72. Recovering Indifference Curves
Now we have upper and lower bounds on where the indifference curve containing bundle A may lie.

73. Recovering Indifference Curvesx2
All bundles revealed to be preferred to A A x1
All bundles revealed to be less preferred to A

74. Recovering Indifference Curvesx2
The region in which the indifference curve containing bundle A must lie. A x1