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. 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 

11. 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 

12. 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 

13. 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.

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

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

16. 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 

17. 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 

18. 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?

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

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

21. Checking if Data Violate the WARP

23. (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. When prices are (2,4), Ms. Consumer chooses the bundle (7,9), and when prices are (15,3), she chooses the bundle (10,3). Is her behavior consistent with the weak axiom of revealed preference? a) Yes b) No c) We would have to observe a third choice to be able to say. d) We can’t tell because we are not told her income in the two cases. e) None of the above.

25. When the prices were (3,1), Vanessa chose the bundle (x,y) = (7,3). Now at the new prices, (p x,p y ), she chooses the bundle (x,y) = (6,5). For Vanessa’s behavior to be consistent with the weak axiom of revealed preference, it must be that: a) 2p y < p x b) p x < 2p y c) 3p y < p x d) p y = 3p x e) None of the above.