chapter 3 preferences. 2 introduction the economic model of consumer behavior is very simple: people...

Chapter 3Preferences

2

Introduction The economic model of consumer behavior is

very simple: people choose the best things they can afford.

The last chapter was devoted to clarifying the meaning of “can afford”.

This chapter will be devoted to clarifying the economic concept of “best things”. A consumer always chooses his most preferred one from his set of available alternatives.

So to model consumers’ choices we must model their preferences.

3

Preference Relations Let x, y are consumption bundles. denotes strict preference: x y

means that bundle x is strictly preferred 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.

~ ~

4

Preference Relations 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.

x y and y x imply x y. x y and (not y x) imply x y.

~ ~

~ ~

5

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.

Bundles are always comparable. Again, if both are true, then they

are indifferent to the individual.

~

~

6

Assumptions about Preference Relations Reflexivity: Any bundle x is always

at least as preferred as itself; i.e.

x x.~

7

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.~ ~ ~

8

Indifference Curves

Take a reference bundle x’. The set of all bundles equally preferred to x’ is the indifference curve (set) containing x’; i.e., the set of all bundles {y: y x’}.

Weakly preferred set: bundles that are weakly preferred to x’. {y: y x’}.~

9

Indifference Curvesxx22

xx11

x”x”

x”’x”’

x’ x’ x” x” x”’ x”’

x’

10

Indifference Curvesxx22

xx11

zz xx yy

x

y

zIf the consumer prefers more to less for each good, all bundles to the northeast of the indifference curve are strictly preferred to x, and all bundles to the southwest of the indifference curve are less preferred to x.

11

Indifference Curvesx2

x1

xAll bundles in I1 arestrictly preferred to all in I2.

y

z

All bundles in I2 are strictly preferred to all in I3.

I1

I2

I3

12

Indifference Curvesx2

x1

I(x’)

x

I(x)

WP(x), the set of bundles weakly preferred to x.

13

Indifference Curvesx2

x1

WP(x), the set of bundles weakly preferred to x.

WP(x) includes I(x).

x

I(x)

14

Indifference Curvesx2

x1

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

x

I(x)

15

Indifference Curves Cannot Intersectxx22

xx11

xxyy

zz

II11

I2From I1, x y. From I2, x z.Therefore y z. However, I1 and I2 represent different levels of preference. Contradiction!

16

Goods 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

Slopes of Indifference Curves

Better

Better

Wors

Wors

ee

Good 2Good 2

Good 1Good 1

Two goodsa negatively sloped indifference curve.

18

Bads

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

Slopes of Indifference Curves

Bette

Bette

rr

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One good and onebad a positively sloped indifference curve.

20

Neutrals If one just do not care about whether

or how much to have a commodity, then the commodity 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

Slopes of Indifference Curves

22

Perfect Substitutes 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.

e.g. orange juice of two different brands.

23

Perfect Substitutesxx22

xx1188

88

1515

1515Slopes are constant at - 1.

I2

I1

Bundles in I2 all have a totalof 15 units and are strictlypreferred to all bundles in I1, which have a total of only 8 units in them.

24

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.

Only the number of pairs of units of the two commodities determines the preference rank-order of bundles.

e.g. left shoes/right shoes.

25

Perfect Complementsxx22

xx11

I1

4545oo

55

99

55 99

Each of (5,5), (5,9) and (9,5) contains5 pairs so each is equally preferred.

26

Perfect Complementsxx22

xx11

I2

I1

4545oo

55

99

55 99

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

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?

28

Indifference Curves Exhibiting Satiationxx22

xx11

SatiationSatiation(bliss)(bliss)pointpoint

29

Indifference Curves Exhibiting Satiationxx22

xx11

BettBett

erer

BettBetterer

Bett

Bett

er

er

SatiationSatiation(bliss)(bliss)pointpoint

30

Indifference Curves Exhibiting Satiationxx22

xx11

BettBett

erer

BettBetterer

Bett

Bett

er

er

SatiationSatiation(bliss)(bliss)pointpoint

31

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.

32

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?

33

Indifference Curves With a Discrete GoodGasoline

Aircraft00 11 22 3 44

Indifference “curves”are collections ofdiscrete points.

34

Well-Behaved Preferences

A preference relation is “well-behaved” if it isMonotonic; and convex.

Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good).

Monotonicity implies that indifference curves are negatively sloped.

35

Well-Behaved Preferences

Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. For example, 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

Well-Behaved Preferences -- Convexity

xx22

yy22

xx22+y+y22

22

xx11 yy11xx11++yy1122

x

y

z = x+y2

z is preferred to both x and y.

37

Well-Behaved Preferences -- Convexity

xx22

yy22

xx11 yy11

x

y

z =(tx1+(1-t)y1, tx2+(1-t)y2) is preferred to x and y

for all 0 < t < 1.

38

Well-Behaved Preferences -- Convexity

xx22

yy22

xx11 yy11

x

y

Preferences are strictly convex when all mixtures z are strictly preferred to their component bundles x and y.

z

39

Weak Convexityx’

y’

z’

Preferences are weakly convex if at least one mixture z is equally preferred to a component bundle.

xz

y

40

Non-Convex Preferences

xx22

yy

22 xx11 yy11

zz

BetterThe mixture zis less preferredthan x or y.

41

More Non-Convex Preferences

xx22

yy

22 xx11 yy11

zz

BetterThe mixture zis less preferredthan x or y.

42

Slopes of Indifference Curves The slope of an indifference curve is

its marginal rate-of-substitution (MRS).

The MRS measures the rate at which the consumer is just willing to substitute one good for the other.

43

Marginal Rate of Substitutionxx22

xx11

x’x’

MRS at x’ is the slope of theindifference curve at x’

44

Marginal Rate of Substitutionxx22

xx11

MRS at x’ isMRS at x’ is lim { lim {xx22//xx11}} xx11 0 0= dx= dx22/dx/dx11 at x’ at x’

xx22

xx11

x’x’

45

Marginal Rate of Substitutionxx22

x1

dxdx22

dxdx11

dxdx22 = MRS = MRS dx dx11 so, at so, at x’, MRS is the rate at x’, MRS is the rate at which the consumer is which the consumer is only just willing to only just willing to exchange commodity 2 exchange commodity 2 for a small amount of for a small amount of commodity 1.commodity 1.

x’x’

46

MRS & Ind. Curve Properties

Better

Better

Wors

Wors

ee

Good 2Good 2

Good 1Good 1

Two goodsTwo goodsa negatively sloped a negatively sloped indifference curveindifference curve

MRS < 0.

47

MRS & Ind. Curve Properties

Bette

Bette

rr

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One good and onebad a positively sloped indifference curve

MRS > 0.

48

MRS & Ind. Curve PropertiesGood 2Good 2

Good 1Good 1

MRS = MRS = - - 55

MRS = MRS = - - 0.50.5

MRS always increases (decreases in absolute value) with x1 (becomes less negative) if and only if preferences are strictly convex.

We call it a diminishing marginal rate of substitution..

49

MRS & Ind. Curve Properties

xx11

xx22MRS = - 0.5

MRS = - 5

MRS decreasesMRS decreases(becomes more (becomes more negative)negative)as xas x11 increases with increases with nonconvex preferencesnonconvex preferences.

50

MRS & Ind. Curve Propertiesxx22

xx11

MRS= - 0.5

MRS = - 1

MRS = - 2

MRS is not always increasing as x1 increases with nonconvex preferences.