chapter three preferences 偏好. structure 3.1 preference relations 3.2 assumptions about...

Chapter Three

Preferences

偏好

Structure

3.1 Preference relations 3.2 Assumptions about preferences 3.3 Graphical representation of preferences

—indifference curve (无差异曲线 ) 3.3.1 definitions and its properties 3.3.2 Slope and marginal rate of substitution (边际替代率 )

3.3.3 Shapes and different preferences

Rationality in Economics

Behavioral Postulate:A decision-maker always chooses its most preferred alternative from its set of available alternatives.

So to model choice we must model decision-makers’ preferences.

3.1 Preference Relations Comparing two different consumption

bundles, x and y: strict preference (严格偏好 ): more preferred (x

y). Indifference (无差异 ): exactly as preferred as

(x~y). weak preference (弱偏好 ): at least as preferred

as (x y).

~

3.2 Assumptions about Preference Relations 1) Completeness 2) Reflexivity 3) Transitivity 4) Monotone 5) Convexity

1) Completeness (完备性 ) For any two bundles x and y it is always

possible to make the statement that either x y or y x

or both.

~

~

2) Reflexivity (反身性 ) Any bundle x is always at least as preferred

as itself; i.e.

x x.~

3) Transitivity (传递性 ) If

x 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. Rational preferences: 1) +3).

~ ~ ~

4) Monotone (单调性 )

Monotone: More of any commodity is always preferred.

The commodities that consumers prefer less of are called bads.

5) Convexity (凸性 )

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

Strict Convexity.

xx22

yy22

xx22+y+y22

22

xx11 yy11xx11+y+y11

22

x

y

z = x+y

2is strictly preferred is strictly preferred to both x and y.to both x and y.

Strict 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.

Strict 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

Weak Convexity.

x’

y’

z’

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

xz

y

Non-Convex Preferences

xx22

yy22

xx11 yy11

zz

Better The mixture zThe mixture zis less preferredis less preferredthan x or y.than x or y.

More Non-Convex Preferences

xx22

yy22

xx11 yy11

zz

BetterThe mixture zThe mixture zis less preferredis less preferredthan x or y.than x or y.

Well-Behaved Preferences

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

3.3 Indifference Curve

3.3.1 Definitions and Properties Indifference curve: a curve representing all

combinations of goods that provide a consumer with the same level of utility.

Indifference Curves

xx22

xx11

x”x”

x”’x”’

x’ x’ x” x” x”’ x”’x’

Indifference Curves

xx22

xx11

zz xx yy

x

y

z

Indifference Curves

x2

x1

x

All 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

Indifference Map

There are many indifference curves that represent different utility levels.

Your satisfaction increases as you move to higher (i.e., to the northeast) indifference curves.

Why?

x2

x1

I(x’)

x

I(x)

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

Weakly Preferred Set (弱偏好集 )

Strictly Preferred Set (严格偏好集 )x2

x1

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

x

I(x)

Proof by contradiction

Movie

CDUU11

UU22AA

CC

BB

Indifference curve cannot intersect

Well-behaved Indifference Curve If preferences are monotonic and (strict)

convex, then the indifference curve that represents the preferences is well-behaved.

Well-behaved Indifference Curve

xx22

xx11

The slope of indifference curve measures the marginal rate of substitution (MRS).

MRS: The amount of good x2 that a consumer would give up to obtain one more unit of good x1 while holding utility constant.

MRS=- x2/x1---is positive

The slope is x2/x1---is negative. Different from the book.

3.3.2 Slope of indifference curve

Marginal Rate of Substitution

xx22

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’

Marginal Rate of Substitution

xx22

x1

dxdx22

dxdx11

dxdx22 = MRS*dx = MRS*dx11 so, at x’, MRS is so, at x’, MRS is

the rate at which the consumer the rate at which the consumer is only is only just willingjust willing to exchange to exchange commodity 2 for a small commodity 2 for a small amount of commodity 1.amount of commodity 1.Marginal willingness to pay.Marginal willingness to pay.

x’x’

Slope and goods

Better

Better

Worse

Worse

Good 2Good 2

Good 1Good 1

Slope < 0.Slope < 0.

Good and bad

Better

Better

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

Slope > 0.Slope > 0.

Movies

CDs

A

D

B

C

Diminishing MRS: There is a decline in the amount of good x2 that the consumer will give up for an additional unit of good x1.

Why diminishing?

Diminishing MRS

3.3.3 The shapes of indifference curves The shape of an indifference curve

describes a consumer‘s willingness to subtitute one good for another.

What would a movie lover’s indifference curves look like?

Movies

CD

U1

U2

U3

Movie Fanatic

Movies

CDU1

U2

U3

CD Fanatic

Extreme Cases of Indifference Curves Perfect substitutes (完全替代 ) Perfect complements (完全互补 ) Satiation (饱和 ) Discrete goods (离散商品 )

Perfect substitutes and perfect complements Perfect substitutes: a consumer is willing to

substitute one good for the other at a constant rate. MRS is (slope =). Total bundles matter.

Perfect complements: Two goods that are always bought in the same ratio.

MRS is . Indifference curves are shaped as right angles.

Perfect Substitutes

xx22

xx1188

88

1515

1515 I2

I1

Bundles in IBundles in I22 all have a total all have a total

of 15 units and are strictlyof 15 units and are strictlypreferred to all bundles inpreferred to all bundles in I I11, which have a total of, which have a total of

only 8 units in them. only 8 units in them.

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 .

The number of pairs determines the preference rank-order of bundles.

Extreme Cases of Indifference Curves; Perfect Complements

xx22

xx11

I1

4545oo

55

99

55 99

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

Extreme Cases of Indifference Curves; Perfect Complements

xx22

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.

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?

Indifference Curves Exhibiting Satiation

xx22

xx11

SatiationSatiation(bliss)(bliss)pointpoint

Indifference Curves Exhibiting Satiation

xx22

xx11

BetteBette

rr

BetteBetterr

Bet

teB

ette

rr

SatiationSatiation(bliss)(bliss)pointpoint

Indifference Curves Exhibiting Satiation

xx22

xx11

BetteBette

rr

BetteBetterr

Bet

teB

ette

rr

SatiationSatiation(bliss)(bliss)pointpoint

Monotone and Satiation

What is the relationship between monotone and Satiation?

Discrete Commodities

A commodity is infinitely divisible if it can be acquired in any quantity; e.g. water or sugar.

A commodity is discrete if it comes in unit lumps of 1, 2, 3, … and so on; e.g. cars, ships and refrigerators.

Indifference Curves for Discrete Commodities Suppose commodity 2 is an infinitely divisible

good (sugar) while commodity 1 is a discrete good (car). What do indifference “curves” look like?

Indifference Curves With a Discrete Good

SugarSugar

CarCar00 11 22 3 44

Indifference “curves”Indifference “curves”are collections ofare collections ofdiscrete points.discrete points.