1. Chapter Three Preferences

2. Rationality in Economics u Behavioral Postulate: An economic decisionmaker always chooses her most preferred alternative from the set of available alternatives. u It is necessary to model the preferences of decisionmakers.

3. Preference Relations u For two consumption bundles, x and y, one of the following holds: –strict preference: x is strictly better than y, or –weak preference: x is at least as good as y, or –indifference: x is exactly as good as is y.

4. Preference Relations u denotes strict preference so x y means that bundle x is preferred strictly to bundle y.   denotes indifference so x  y means that x and y are equally preferred.  denotes weak preference so x y means that bundle x is preferred at least as much as is bundle y. ~  ~   

5. Preference Relations  x y and y x imply x  y. u x y and (not y x) imply x y. ~  ~  ~  ~  

6. Assumptions about Preference Relations u Completeness: It is always possible to rank-order any two bundles x and y by preference; i.e. it is always possible to make the statement that either x y or y x. ~  ~ 

7. Assumptions about Preference Relations u Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x. ~ 

8. Assumptions about Preference Relations u Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z. ~  ~  ~ 

9. Indifference Curves  Take some reference bundle x’. The set of all bundles equally preferred to x’ is called the indifference curve containing x’; i.e. the set of all bundles y for which y  x’. u Since an indifference “curve” is not always a curve a better name might be an indifference “set”.

10. Indifference Curves x2x2x2x2 x1x1x1x1 x” x”’ x’, x” and x”’ are all equally preferred; x’  x”  x”’. x’

11. Indifference Curves x2x2x2x2 x1x1x1x1 zxy z x y  x y z

12. Indifference Curves x2x2 x1x1 x All bundles in I 1 are strictly preferred to all in I 2. y z All bundles in I 2 are strictly preferred to all in I 3. I1I1 I2I2 I3I3

13. Indifference Curves x2x2 x1x1 WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x). x I(x)

14. Indifference Curves x2x2 x1x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x)

15. Indifference Curves Cannot Intersect x2x2x2x2 x1x1x1x1 x y z I1I1I1I1 I2I2 From I 1, x  y. From I 2, x  z. Therefore y  z. But from I 1 and I 2 we see y z, a contradiction. 

16. Slopes of Indifference Curves u Suppose more of commodity 1 is always preferred and that more of commodity 2 is always preferred. u Then both commodities are always goods and indifference curves are negatively sloped.

17. Slopes of Indifference Curves Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve.

18. Slopes of Indifference Curves u Suppose more of commodity 1 is always preferred and that less of commodity 2 is always preferred. u Then commodity 1 is a good, commodity 2 is a bad and indifference curves are positively sloped.

19. Slopes of Indifference Curves Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve.

20. Extreme Cases of Indifference Curves; Perfect Substitutes x2x2x2x2 x1x1x1x1 8 8 15 15 Slopes are constant at - 1. I2I2 I1I1 Bundles in I 2 all have a total of 15 units and are strictly preferred to all bundles in I 1, which have a total of only 8 units in them.

21. Extreme Cases of Indifference Curves; Perfect Complements x2x2x2x2 x1x1x1x1 I2I2 I1I1 45 o 5 9 59 Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less preferred than the bundle (9,9) which contains 9 pairs.

22. Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Satiation (bliss) point

23. Indifference Curves Exhibiting Satiation x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point

24. Well-Behaved Preferences u A preference relation is “well- behaved” if it is –monotonic and convex. u “Monotonic” means that more of any commodity is always preferred (i.e. no satiation and every unit of every commodity is always a good).

25. Well-Behaved Preferences u Convexity means that ‘mixtures’ of bundles are (at least weakly) preferred to the bundles themselves. For example, take two bundles x and y and then construct the 50-50 mixture z = (0.5)x + (0.5)y. Then the bundle z is at least as preferred as either x or y.

26. Well-Behaved Preferences -- Convexity. x2x2x2x2 y2y2y2y2 x 2 +y 2 2 x1x1x1x1 y1y1y1y1 x 1 +y 1 2 x y z = x+y 2 is strictly preferred to both x and y.

27. Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

28. More Non-Convex Preferences x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y.

29. Marginal Rate of Substitution x2x2x2x2 x1x1x1x1 x’ MRS at x’ is the slope of the indifference curve at the point x’ point x’

30. Marginal Rate of Substitution x2x2x2x2 x1x1 dx 2 dx 1 MRS is the rate at which the consumer is just willing to exchange a small amount of commodity 2 ( dx 2 ) for a small amount of commodity 1 ( dx 1 ). x’

31. MRS & Ind. Curve Properties Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve MRS < 0.

32. MRS & Ind. Curve Properties Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS > 0.

33. MRS & Ind. Curve Properties Good 2 Good 1 MRS = - 5 MRS = - 0.5 MRS always increases with x 1 (becomes less negative) if and only if preferences are strictly convex.