1. Course: Microeconomics Text: Varian’s Intermediate Microeconomics 1

2.  Last chapter we talk about what is affordable or feasible to consumers.  This time we talk about preferences : what the consumer like more and what they like less.  As a rational agent, a consumer chooses the option in the budget set that is highest in their preference order (i.e. one likes the most). 2

3.  Let x, y are consumption bundles.  denotes strict preference so x y means that bundle x is preferred strictly to bundle y.   denotes 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. ~    ~  3

4.  Strict preference, weak preference and indifference are all preference relations.  Particularly, they are ordinal relations; i.e. they state only the order in which bundles are preferred.  It has no indication of how much they like one versus the other. 4

5.  x y and y x imply x  y.  x y and (not y x) imply x y. ~  ~  ~  ~   5

6.  Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x.  Bundles are always comparable.  If both are true, then they are indifferent to the individual. ~  ~  6

7.  Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x. ~  7

8.  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.  It avoids circular preference, and ensure that there exists the best bundle. ~  ~  ~  8

9.  Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve (set) containing x’; the set of all bundles {y: y  x’}.  Weakly preferred set: bundles that are weakly preferred to x’. {y: y x’}. ~  9

10. x2x2x2x2 x1x1x1x1 x” x”’ x’  x”  x”’ x’ The consumer is indifferent between every point on the indifference curve. 10

11. x2x2x2x2 x1x1x1x1 zxy z x y  x y z If consumer prefers more to less for each goods, all bundles on the northeast of the indifference curve are strictly preferred to x, and all bundles southwest of the indifference curve are less preferred to x. 11

12. 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 12

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

14. x2x2 x1x1 WP(x), the set of bundles weakly preferred to x. WP(x) includes I(x). x I(x) 14

15. x2x2 x1x1 SP(x), the set of bundles strictly preferred to x, does not include I(x). x I(x) 15

16. x2x2x2x2 x1x1x1x1 x y z I1I1I1I1 I2I2 From I 1, x  y. From I 2, x  z. Therefore y  z. But because I 1 and I 2 represent distinct level of preference, we see y z, a contradiction.  16

17.  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.  It is because when one has more of one good, one has to get less of another to make this bundle indifferent to the original one. 17

18. Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve. 18

19.  If less of a commodity is always preferred then the commodity is a bad.  e.g. rotten fruits; tobacco smoke (if you do not smoke)  If one good is good and the other is bad, then the indifference curve would be upward sloping. 19

20. Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve. If you want more of the good, you also have to get more of the bad so that you are indifferent between them. 20

21.  If one just do not care about whether or how much to have a commodity, this is called a neutral good.  E.g.: goods that you don’t use and do not care about their existence.  If one commodity is neutral, the other is good, the indifference curve would be vertical / horizontal. 21

22. 22

23.  If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes.  Only the total amount (or a weighted sum) of the two commodities in bundles determines their preference rank-order.  Example: orange juice of two different brands. Apartment in different locations. 23

24. 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. 24

25.  If a consumer always consumes commodities 1 and 2 in fixed proportion (e.g. one-to-one), then the commodities are perfect complements.  Only the number of pairs of units of the two commodities determines the preference rank-order of bundles.  E.g.: left shoes/right shoes; computer and monitor. 25

26. x2x2x2x2 x1x1x1x1 I1I1 45 o 5 9 59 Each of (5,5), (5,9) and (9,5) contains 5 pairs so each is equally preferred. 26

27. 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. 27

28.  A bundle strictly preferred to any others is a satiation point or a bliss point.  The satiation point is the best bundle. More of anything are not better.  What do indifference curves look like for preferences exhibiting satiation? 28

29. x2x2x2x2 x1x1x1x1 Satiation (bliss) point 29

30. x2x2x2x2 x1x1x1x1 Better Better Better Satiation (bliss) point 30

31. x2x2x2x2 x1x1x1x1 Bette r Satiation (bliss) point 31

32.  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. 32

33.  Suppose commodity 2 is an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like? 33

34. Gasoline Aircraft 01234 Indifference “curves” are collections of discrete points. 34

35.  Typical assumptions of preferences  A preference relation is “well-behaved” if it is  monotonic and convex.  Monotonicity: More of any commodity is always preferred ( i.e. no satiation and every commodity is a good).  This implies a negatively sloped IC. 35

36.  Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 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. 36

37. x2x2x2x2 y2y2y2y2 x 2 +y 2 2 x1x1x1x1 y1y1y1y1 x 1 +y 1 2 x y z = x+y 2 Is (weakly) preferred to both x and y. 37

38. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 x y z =(tx 1 +(1-t)y 1, tx 2 +(1-t)y 2 ) is preferred to x and y for all 0 < t < 1. 38

39. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 x y Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y. z 39

40. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y. One likes both, but does want to consume together. 40

41. x2x2x2x2 y2y2y2y2 x1x1x1x1 y1y1y1y1 z Better The mixture z is less preferred than x or y. 41

42.  It represents a more balanced preference rather than a preference that induces a specialization.  It is natural to consume the goods involved in positive amount.  It also implies a diminishing marginal rate of substitution. 42

43.  The negative of the slope of an indifference curve is its marginal rate of substitution (MRS).  Note: this is slightly different from the textbook, but my definition is more popular.  This represents the maximum amount of x 2 one is willing to give up per unit of x 1 at a specific consumption bundle. 43

44. x2x2x2x2 x1x1x1x1 x’ MRS at x’ is the (negative) slope of the indifference curve at x’ 44

45. x2x2x2x2 x1x1x1x1 MRS at x’ is lim -{x 2 /x 1 } x 1 0 = -dx 2 /dx 1 at x’ MRS at x’ is lim -{  x 2 /  x 1 }  x 1 0 = -dx 2 /dx 1 at x’ x2x2x2x2 x1x1x1x1 x’ 45

46. x2x2x2x2 x1x1 dx 2 dx 1 -dx 2 = MRS dx 1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. -dx 2 = MRS  dx 1 so, at x’, MRS is the rate at which the consumer is only just willing to exchange commodity 2 for a small amount of commodity 1. x’ 46

47. Better Worse Good 2 Good 1 Two goods a negatively sloped indifference curve MRS > 0. 47

48. Better Worse Good 2 Bad 1 One good and one bad a positively sloped indifference curve MRS < 0. Because instead of giving up, you have to obtain more good 2 for you to be willing to accept more good 1. 48

49. Good 2 Good 1 MRS = 5 MRS = 0.5 MRS always decreases with x 1 if and only if preferences are strictly convex. We call it a diminishing marginal rate of substitution. 49

50. x1x1x1x1 x2x2x2x2 MRS = 0.5 MRS = 5 MRS increases as x 1 increases non-convex preferences 50

51. x2x2x2x2 x1x1x1x1 MRS = 0.5 MRS = 2 MRS is not always decreasing as x 1 increases non-convex preferences. MRS=1.5 51

52.  In this chapter, we talk about how we can specify consumer’s preference towards different consumption bundles.  We can use indifferent curve to depict different kinds of preferences.  The marginal rate of substitution is the slope of indifference curve. It represents the willingness to substitute one good for another one. 52

53.  We have talked about preference and indifference curve in this chapter.  To put preference in a more mathematically convenient way, we introduce the utility function in the coming chapter.  Then we can put together preference/utility and budget constraint to analyze consumer choices. 53