Preferences W. W. Norton & Company, Inc.

Transcription

1 Preferences 2010 W. W. Norton & Company, Inc.

2 Rationality in Economics Behavioral Postulate: A decisionmaker always chooses its most preferred alternative from its set of available alternatives. So to model choice we must model decisionmakers preferences W. W. Norton & Company, Inc. 2

3 Preference Relations Comparing two different consumption bundles, x and y: strict preference: x is more preferred than is y. weak preference: x is as at least as preferred as is y. indifference: x is exactly as preferred as is y W. W. Norton & Company, Inc. 3

4 Preference Relations Strict preference, weak preference and indifference are all preference relations. Particularly, l they are ordinal relations; i.e. they state only the order in which bundles are preferred W. W. Norton & Company, Inc. 4

5 Preference Relations denotes strict preference; x y means that bundle x is preferred strictly to bundle y W. W. Norton & Company, Inc. 5

6 Preference Relations denotes strict preference; x y means bundle x is preferred strictly to bundle y. denotes d indifference; x y means x and y are equally preferred W. W. Norton & Company, Inc. 6

7 Preference Relations denotes strict preference so x y means that bundle x is preferred strictly to bundle y. denotes d indifference; x y means x and y are equally preferred. ~ denotes weak preference; x y means x is preferred at least as ~ much as is y W. W. Norton & Company, Inc. 7

8 Preference Relations x y and y x imply x y. ~ ~ 2010 W. W. Norton & Company, Inc. 8

9 Preference Relations x y and y x imply x y. ~ ~ x y and (not y x) imply x y. ~ ~ 2010 W. W. Norton & Company, Inc. 9

10 Assumptions about Preference Relations Completeness: For any two bundles x and y it is always possible to make the statement that either x y ~ or y x. ~ 2010 W. W. Norton & Company, Inc. 10

11 Assumptions about Preference Relations Reflexivity: Any bundle x is always at least as preferred as itself; ie i.e. x x. ~ 2010 W. W. Norton & Company, Inc. 11

12 Assumptions about Preference Relations 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; ie i.e. x y and y z x z. ~ ~ ~ 2010 W. W. Norton & Company, Inc. 12

13 Indifference Curves Take a reference bundle x. The set of all bundles equally preferred to x is the indifference curve containing x ; the set of all bundles y x. Since an indifference curve is not always a curve a better name might be an indifference set W. W. Norton & Company, Inc. 13

14 Indifference Curves x 2 x x x x x x x W. W. Norton & Company, Inc. 14

15 Indifference Curves x 2 z x y x z y x W. W. Norton & Company, Inc. 15

16 Indifference Curves x 2 x I 2 I 1 z All bundles in I 1 are strictly preferred to all in I 2. y I 3 All bundles in I 2 are strictly preferred to all in I 3. x W. W. Norton & Company, Inc. 16

17 Indifference Curves x 2 x WP(x), the set of bundles weakly preferred to x. I(x) I(x ) x W. W. Norton & Company, Inc. 17

18 Indifference Curves x 2 x WP(x), the set of bundles weakly preferred to x. I(x) WP(x) includes I(x). x W. W. Norton & Company, Inc. 18

19 Indifference Curves x 2 x SP(x), the set of bundles strictly preferred to x, I(x) does not include I(x). x W. W. Norton & Company, Inc. 19

20 Indifference Curves Cannot Intersect x 2 I 1 I 2 From I 1, x y. From I 2, x z. Therefore y z. x z y x W. W. Norton & Company, Inc. 20

21 Indifference Curves Cannot Intersect x 2 I 1 I 2 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. ct o x y z x W. W. Norton & Company, Inc. 21

22 Slopes of Indifference Curves When more of a commodity is always preferred, the commodity is a good. If every commodity is a good then indifference curves are negatively sloped W. W. Norton & Company, Inc. 22

23 Slopes of Indifference Curves Good 2 Two goods a negatively sloped indifference curve. Good W. W. Norton & Company, Inc. 23

24 Slopes of Indifference Curves If less of a commodity is always preferred then the commodity is a bad W. W. Norton & Company, Inc. 24

25 Slopes of Indifference Curves Good 2 One good and one bad a positively sloped indifference curve. Bad W. W. Norton & Company, Inc. 25

26 Extreme Cases of Indifference Curves; Perfect Substitutes If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes and only the total amount of the two commodities in bundles determines their preference rank-order W. W. Norton & Company, Inc. 26

27 Extreme Cases of Indifference Curves; Perfect Substitutes x 2 15 I 2 Slopes are constant at - 1. Bundles in I 2 all have a total 8 of 15 units and are strictly preferred pee edtoa all bundles des in I 1 I 1, which have a total of only 8 units in them x W. W. Norton & Company, Inc. 27

28 Extreme Cases of Indifference Curves; Perfect Complements If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements and only the number of pairs of units of the two commodities determines the preference rank-order of bundles W. W. Norton & Company, Inc. 28

29 Extreme Cases of Indifference Curves; Perfect Complements x o Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred. I x W. W. Norton & Company, Inc. 29

30 Extreme Cases of Indifference Curves; Perfect Complements x o Since each of (5,5), (5,9) and (9,5) contains 5 pairs, each is less I 2 preferred than the bundle (9,9) 9) which I 1 contains 9 pairs. 5 9 x W. W. Norton & Company, Inc. 30

31 Preferences Exhibiting Satiation A bundle strictly preferred to any other is a satiation point or a bliss point. What do indifference curves look like for preferences exhibiting satiation? 2010 W. W. Norton & Company, Inc. 31

32 Indifference Curves Exhibiting Satiation x 2 Satiation (bliss) point x W. W. Norton & Company, Inc. 32

33 Indifference Curves Exhibiting Satiation x 2 Satiation (bliss) point Bette er x W. W. Norton & Company, Inc. 33

34 Indifference Curves Exhibiting Satiation x 2 Satiation (bliss) point Bette er x W. W. Norton & Company, Inc. 34

35 Indifference Curves for Discrete Commodities A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or cheese. A commodity is discrete if it comes in unit lumps of 1, 2, 3, and so on; e.g. aircraft, ships and refrigerators W. W. Norton & Company, Inc. 35

36 Indifference Curves for Discrete Commodities Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference curves look like? 2010 W. W. Norton & Company, Inc. 36

37 Indifference Curves With a Discrete Good Gas- oline Indifference curves are collections of discrete points Aircraft 2010 W. W. Norton & Company, Inc. 37

38 Well-Behaved Preferences A preference relation is wellbehaved if it is monotonic and convex. Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good) W. W. Norton & Company, Inc. 38

39 Well-Behaved Preferences Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the mixture of the bundles x and y is z = (0.5)x + (0.5)y. z is at least as preferred as x or y W. W. Norton & Company, Inc. 39

40 Well-Behaved Preferences -- Convexity. x 2 x 2 +y 2 2 y 2 x x+y z = 2 y x 1 x 1 +y y is strictly preferred to both x and y W. W. Norton & Company, Inc. 40

41 Well-Behaved Preferences -- Convexity. x 2 y 2 x z =(tx 1+(1-t)y 1, tx 2+(1-t)y 2) is preferred to x and y for all 0 < t < 1. y x 1 y W. W. Norton & Company, Inc. 41

42 Well-Behaved Preferences -- Convexity. x 2 y 2 x Preferences are strictly convex when all mixtures z are strictly z preferred to their component bundles x and y. y x 1 y W. W. Norton & Company, Inc. 42

43 Well-Behaved Preferences -- Weak Convexity. x z x z y y Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle W. W. Norton & Company, Inc. 43

44 Non-Convex Preferences x 2 z The mixture z is less preferred than x or y. y 2 x 1 y W. W. Norton & Company, Inc. 44

45 More Non-Convex Preferences x 2 z The mixture z is less preferred than x or y. y 2 x 1 y W. W. Norton & Company, Inc. 45

46 Slopes of Indifference Curves The slope of an indifference curve is its marginal rate-of-substitution (MRS). How can a MRS be calculated? l 2010 W. W. Norton & Company, Inc. 46

47 Marginal Rate of Substitution x 2 MRS at x is the slope of the indifference curve at x x x W. W. Norton & Company, Inc. 47

48 Marginal Rate of Substitution x 2 x 2 x MRS at x is lim { x 2 / x 1 } x 1 0 = dx 2 /dx 1 at x x 1 x W. W. Norton & Company, Inc. 48

49 x 2 Marginal Rate of Substitution dx 2 dx 1 x dx 2 = MRS dx 1 so, at x, MRS is the rate at which the consumer is only yj just willing to exchange commodity 2 for a small amount of commodity 1. x W. W. Norton & Company, Inc. 49

50 MRS & Ind. Curve Properties Good 2 Two goods a negatively sloped indifference curve MRS < 0. Good W. W. Norton & Company, Inc. 50

51 MRS & Ind. Curve Properties Good 2 One good and one bad a positively sloped indifference curve MRS > 0. Bad W. W. Norton & Company, Inc. 51

52 MRS & Ind. Curve Properties Good 2 MRS = -5 MRS always increases with x 1 (becomes less negative) if and only if preferences are strictly convex. MRS = Good W. W. Norton & Company, Inc. 52

53 MRS & Ind. Curve Properties x 2 MRS = MRS decreases (becomes more negative) as x 1 increases nonconvex preferences MRS = - 5 x W. W. Norton & Company, Inc. 53

54 MRS & Ind. Curve Properties x 2 MRS = MRS is not always increasing as x 1 increases nonconvex preferences. MRS = - 1 MRS = - 2 x W. W. Norton & Company, Inc. 54