1. Chapter Seven Revealed Preference

2. Revealed Preference Analysis u 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...

3. Revealed Preference Analysis –Test the behavioral hypothesis that a consumer chooses the most preferred bundle from those available. –Discover the consumer’s preference relation.

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

5. Assumptions on Preferences x2x2 x1x1 x1*x1* x2*x2* If preferences are convex and monotonic (i.e. well-behaved) then the most preferred affordable bundle is unique.

6. Direct Preference Revelation u 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 x2x2 x1x1 x*x* y The chosen bundle x * is revealed directly as preferred to the bundles y and z. z

8. Direct Preference Revelation u That x is revealed directly as preferred to y will be written as x y. D 

9. Indirect Preference Revelation u 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 z so x y and y z x z. D  D  I  I 

10. Indirect Preference Revelation x2x2 x1x1 x* z z is not affordable when x* is chosen.

11. Indirect Preference Revelation x2x2 x1x1 x* y* z x* is not affordable when y* is chosen.

12. Indirect Preference Revelation x2x2 x1x1 x* y* z z is not affordable when x* is chosen. x* is not affordable when y* is chosen.

13. z is not affordable when x* is chosen. x* is not affordable when y* is chosen. So x* and z cannot be compared directly. Indirect Preference Revelation x2x2 x1x1 x* y* z

14. But x*x* y* z is not affordable when x* is chosen. x* is not affordable when y* is chosen. So x* and z cannot be compared directly. Indirect Preference Revelation x2x2 x1x1 x* y* z D 

15. But x*x* y* and y* z z is not affordable when x* is chosen. x* is not affordable when y* is chosen. So x* and z cannot be compared directly. Indirect Preference Revelation x2x2 x1x1 x* y* z D  D 

16. z is not affordable when x* is chosen. x* is not affordable when y* is chosen. So x* and z cannot be compared directly. Indirect Preference Revelation x2x2 x1x1 x* y* z But x*x* y* and y* z so x* z. D  D  I 

17. Two Axioms of Revealed Preference u 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) u 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). D  D 

19. The Weak Axiom of Revealed Preference (WARP) u Choice data which violate the WARP are inconsistent with economic rationality. u The WARP is a necessary condition for applying economic rationality to explain observed choices.

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

21. The Weak Axiom of Revealed Preference (WARP) x2x2 x1x1 x y

22. x2x2 x1x1 x y x is chosen when y is available so x y. D 

23. The Weak Axiom of Revealed Preference (WARP) x2x2 x1x1 x y y is chosen when x is available so y x. x is chosen when y is available so x y. D  D 

24. The Weak Axiom of Revealed Preference (WARP) x2x2 x1x1 x y These statements are inconsistent with each other. x is chosen when y is available so x y. y is chosen when x is available so y x. D  D 

25. Checking if Data Violate the WARP u A consumer makes the following choices: –At prices (p 1,p 2 )=($2,$2) the choice was (x 1,x 2 ) = (10,1). –At (p 1,p 2 )=($2,$1) the choice was (x 1,x 2 ) = (5,5). –At (p 1,p 2 )=($1,$2) the choice was (x 1,x 2 ) = (5,4). u Is the WARP violated by these data?

26. Checking if Data Violate the WARP

27. Red numbers are costs of chosen bundles.

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

29. Checking if Data Violate the WARP

31. (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.

32. Checking if Data Violate the WARP (5,4) (10,1) (10,1) (5,4) x1x1 x2x2 D  D 

33. The Strong Axiom of Revealed Preference (SARP)  If the bundle x is revealed (directly or indirectly) as preferred to the bundle y and x  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 ). D  D  I  I 

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

35. 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.

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

37. Recovering Indifference Curves u Suppose we observe: A: (p 1,p 2 ) = ($1,$1) & (x 1,x 2 ) = (15,15) B: (p 1,p 2 ) = ($2,$1) & (x 1,x 2 ) = (10,20) C: (p 1,p 2 ) = ($1,$2) & (x 1,x 2 ) = (20,10) D: (p 1,p 2 ) = ($2,$5) & (x 1,x 2 ) = (30,12) E: (p 1,p 2 ) = ($5,$2) & (x 1,x 2 ) = (12,30). u Where lies the indifference curve containing the bundle A = (15,15)?

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

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

40. Recovering Indifference Curves u 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:

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

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

43. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10) D: (p 1,p 2 )=(2,5); (x 1,x 2 )=(30,12) E: (p 1,p 2 )=(5,2); (x 1,x 2 )=(12,30).

44. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10) D: (p 1,p 2 )=(2,5); (x 1,x 2 )=(30,12) E: (p 1,p 2 )=(5,2); (x 1,x 2 )=(12,30). Begin with bundles revealed to be less preferred than bundle A.

45. Recovering Indifference Curves x2x2 x1x1 A A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15).

46. Recovering Indifference Curves x2x2 x1x1 A A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15).

47. Recovering Indifference Curves x2x2 x1x1 A A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15). A is directly revealed preferred to any bundle in

48. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20).

49. Recovering Indifference Curves x2x2 x1x1 A B A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20).

50. Recovering Indifference Curves x2x2 x1x1 A B A is directly revealed preferred to B and …

51. Recovering Indifference Curves x2x2 x1x1 B B is directly revealed preferred to all bundles in

52. Recovering Indifference Curves x2x2 x1x1 B so, by transitivity, A is indirectly revealed preferred to all bundles in

53. Recovering Indifference Curves x2x2 x1x1 B so A is now revealed preferred to all bundles in the union. A

54. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10).

55. Recovering Indifference Curves x2x2 x1x1 AC A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10).

56. Recovering Indifference Curves x2x2 x1x1 AC A is directly revealed preferred to C and...

57. Recovering Indifference Curves x2x2 x1x1 C C is directly revealed preferred to all bundles in

58. Recovering Indifference Curves x2x2 x1x1 C so, by transitivity, A is indirectly revealed preferred to all bundles in

59. Recovering Indifference Curves x2x2 x1x1 C so A is now revealed preferred to all bundles in the union. B A

60. Recovering Indifference Curves x2x2 x1x1 C so A is now revealed preferred to all bundles in the union. B A Therefore the indifference curve containing A must lie everywhere else above this shaded set.

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

62. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10) D: (p 1,p 2 )=(2,5); (x 1,x 2 )=(30,12) E: (p 1,p 2 )=(5,2); (x 1,x 2 )=(12,30). A

63. Recovering Indifference Curves x2x2 x1x1 D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) D: (p 1,p 2 )=(2,5); (x 1,x 2 )=(30,12). A

64. Recovering Indifference Curves x2x2 x1x1 D D is directly revealed preferred to A. A

65. Recovering Indifference Curves x2x2 x1x1 D D is directly revealed preferred to A. Well-behaved preferences are convex A

66. Recovering Indifference Curves x2x2 x1x1 D 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

67. Recovering Indifference Curves x2x2 x1x1 D 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 As well,...

68. Recovering Indifference Curves x2x2 x1x1 D 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

69. Recovering Indifference Curves x2x2 x1x1 D A bundles revealed to be strictly preferred to A

70. Recovering Indifference Curves x2x2 x1x1 A B E C D A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) B: (p 1,p 2 )=(2,1); (x 1,x 2 )=(10,20) C: (p 1,p 2 )=(1,2); (x 1,x 2 )=(20,10) D: (p 1,p 2 )=(2,5); (x 1,x 2 )=(30,12) E: (p 1,p 2 )=(5,2); (x 1,x 2 )=(12,30). A

71. Recovering Indifference Curves x2x2 x1x1 A E A: (p 1,p 2 )=(1,1); (x 1,x 2 )=(15,15) E: (p 1,p 2 )=(5,2); (x 1,x 2 )=(12,30).

72. Recovering Indifference Curves x2x2 x1x1 A E E is directly revealed preferred to A.

73. Recovering Indifference Curves x2x2 x1x1 A E E is directly revealed preferred to A. Well-behaved preferences are convex

74. Recovering Indifference Curves x2x2 x1x1 A E 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.

75. Recovering Indifference Curves x2x2 x1x1 A E 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. As well,...

76. Recovering Indifference Curves x2x2 x1x1 A E 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.

77. Recovering Indifference Curves x2x2 x1x1 A E More bundles revealed to be strictly preferred to A

78. Recovering Indifference Curves x2x2 x1x1 A B C E D Bundles revealed earlier as preferred to A

79. Recovering Indifference Curves x2x2 x1x1 B C E D All bundles revealed to be preferred to A A

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

81. Recovering Indifference Curves x2x2 x1x1 All bundles revealed to be preferred to A A All bundles revealed to be less preferred to A

82. Recovering Indifference Curves x2x2 x1x1 All bundles revealed to be preferred to A A All bundles revealed to be less preferred to A

83. Recovering Indifference Curves x2x2 x1x1 The region in which the indifference curve containing bundle A must lie. A

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

85. Index Numbers u Two basic types of indices –price indices, and –quantity indices u Each index compares expenditures in a base period and in a current period by taking the ratio of expenditures.