Revealed Preference

1. 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 • Preferences • do not change while the choice data are gathered. • are strictly convex. • are monotonic.

4. Assumptions on Preferences x2 If preferences are strictly convex and monotonic then the most preferredaffordable 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 Revelation x2 The chosen bundle x* isrevealed directly as preferredto 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 zso x DRP y and y DRP z x IRP z.

9. Indirect Preference Revelation x2 z is not affordable when x* is chosen. x* z x1

10. Indirect Preference Revelation x2 x* is not affordable when y* is chosen. x* y* z x1

11. 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

12. 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*and y*DRPz 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) x2 x is chosen when y is availableso x DRP y. y x x1

18. 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

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 thatwere not chosen.

22. Checking if Data Violate the WARP

23. 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.

24. Checking if Data Violate the WARP x2 (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 A: ($1,$3,$10) (3,1,4). 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 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 ...

31. 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 ...

32. The Strong Axiom of Revealed Preference The data do not violatethe WARP but there are3 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 WARPis 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; neitherWARP 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 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.

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

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

44. 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

45. 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

46. Recovering Indifference Curves x2 A is directly revealed preferred to B and … B A x1

47. Recovering Indifference Curves x2 B is directly revealed preferredto all bundles in B x1

48. Recovering Indifference Curves x2 so, by transitivity, A is indirectlyrevealed preferred to all bundles in B x1

49. Recovering Indifference Curves x2 so A is now revealed preferredto all bundles in the union. B A x1

50. 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