Chapter Three

1. Chapter Three Preferences

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

3. Preference Relations • 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. f f ~ ~ Preference Relations p • 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 sox y means that bundle x is preferred at least as much as is bundle y. p

5. f f f f ~ ~ ~ ~ Preference Relations • x y and y x imply x ~ y. • x y and (not y x) imply x y. p

6. f f ~ ~ Assumptions about Preference Relations • 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. f ~ Assumptions about Preference Relations • Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.

8. f f f ~ ~ ~ Assumptions about Preference Relations • Transitivity: Ifx is at least as preferred as y, andy is at least as preferred as z, thenx 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’. • Since an indifference “curve” is not always a curve a better name might be an indifference “set”.

10. Indifference Curves x2 x’, x” and x”’ are all equally preferred;x’ ~ x” ~ x”’. x’ x” x”’ x1

11. Indifference Curves x2 zxy p p x z y x1

12. Indifference Curves I1 All bundles in I1 are strictly preferred to all in I2. x2 x z I2 All bundles in I2 are strictly preferred to all in I3. y I3 x1

13. Indifference Curves x2 WP(x), the set of bundles weakly preferred to x. x WP(x) includes I(x). I(x) x1

14. Indifference Curves x2 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) x1

15. Indifference Curves Cannot Intersect From I1, x ~ y. From I2, x ~ z. Therefore y ~ z. But from I1 and I2 we see y z, a contradiction. I2 x2 I1 p x y z x1

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

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

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

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

20. Extreme Cases of Indifference Curves; Perfect Substitutes x2 Slopes are constant at - 1. 15 • I2 Bundles in I2 all have a totalof 15 units and are strictly preferred to all bundles in I1, which have a total of only 8 units in them. 8 I1 x1 8 15

21. Extreme Cases of Indifference Curves; Perfect Complements x2 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. 45o 9 I2 5 I1 x1 5 9

22. Indifference Curves Exhibiting Satiation x2 Satiation(bliss)point x1

23. Indifference Curves Exhibiting Satiation x2 Better Better Satiation(bliss)point Better x1

24. Well-Behaved Preferences • A preference relation is “well-behaved” if it is • monotonic and convex. • “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 • 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. x x2 x+y is strictly preferred to both x and y. x2+y2 z = 2 2 y y2 x1+y1 x1 y1 2

27. Non-Convex Preferences x2 Better The mixture zis less preferred than x or y. z y2 x1 y1

28. More Non-Convex Preferences x2 Better The mixture zis less preferred than x or y. z y2 x1 y1

29. Marginal Rate of Substitution x2 MRS at x’ is the slope of theindifference curve at the point x’ x’ x1

30. Marginal Rate of Substitution MRS is the rate at which the consumer is just willing to exchange a small amount of commodity 2 (dx2) for a small amount of commodity 1 (dx1). x2 x’ dx2 dx1 x1

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

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

33. MRS & Ind. Curve Properties Good 2 MRS = - 5 MRS always increases with x1 (becomes less negative) if and only if preferences are strictly convex. MRS = - 0.5 Good 1