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Chapter Seven

Chapter Seven. Revealed Preference. 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 . Revealed Preference Analysis.

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Chapter Seven

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  1. Chapter Seven Revealed Preference

  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 ...

  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 • 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. p D Direct Preference Revelation • That x is revealed directly as preferred to y will be written as x y.

  9. p p D D 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 zso x y and y z x z. p I p I

  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. p D 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*x*y* y* z x1

  15. p p D D 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*x*y*and y* z y* z x1

  16. p p D D 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*x*y*and y* z so x* z. y* z p x1 I

  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. p p D D 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 y not (y 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. p D The Weak Axiom of Revealed Preference (WARP) x2 x is chosen when y is availableso x y. y x x1

  23. p p D D The Weak Axiom of Revealed Preference (WARP) x2 x is chosen when y is availableso x y. y is chosen when x is availableso y x. y x x1

  24. p p D D The Weak Axiom of Revealed Preference (WARP) x2 x is chosen when y is availableso x y. y is chosen when x is availableso y 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. p p D D Checking if Data Violate the WARP x2 (5,4)(10,1) (10,1)(5,4) x1

  35. p p D D p I 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 y or x y not ( y x or y x ). p I

  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 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)

  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. p D The Strong Axiom of Revealed Preference In situation A,bundle A is directly revealedpreferred tobundle C; A C.

  41. p D The Strong Axiom of Revealed Preference In situation B,bundle B is directly revealedpreferred tobundle A; B A.

  42. p D The Strong Axiom of Revealed Preference In situation C,bundle C is directly revealedpreferred tobundle B; C B.

  43. The Strong Axiom of Revealed Preference

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

  45. p p p D D D p p p I I I The Strong Axiom of Revealed Preference We have thatA C, B A and C Bso, by transitivity,A B, B C and C A. The data do not violate the WARP but ...

  46. p p p D D D p p p I I I The Strong Axiom of Revealed Preference We have thatA C, B A and C Bso, by transitivity,A B, B C and C A. I I I The data do not violate the WARP but ...

  47. p D p I The Strong Axiom of Revealed Preference B A is inconsistentwith A B. I I I The data do not violate the WARP but ...

  48. p D p I The Strong Axiom of Revealed Preference A C is inconsistentwith C A. I I I The data do not violate the WARP but ...

  49. p D p I The Strong Axiom of Revealed Preference C B is inconsistentwith B 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

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