Ch03 Consumer Preferences

8/10/2019 Ch03 Consumer Preferences

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Chapter 3

Preferences


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Rationality in Economics

The Optimization (Rational Behavior)

Principle:

A decision maker always chooseshis/her most preferred alternative

from the set of available alternatives.

So to model choice we must model

decision-makers

preferences.


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Preference Relations

Comparing two different consumptionbundles, 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 preferredas is y.


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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.


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denotes strict preference so

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.

~

~


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x y and y x imply x y.

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

~ ~

~ ~


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Assumptions about Preference

Relations

Completeness: Any two bundles x

and y can be compared. That is,

either

x y

or

y x

or both holds.

~

~


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Relations

Reflexivity: Any bundle x is always

at least as good as itself;

x x.~


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Relations

Transitivity: If x is at least as good

as y, and y is at least as good as z,

then x is at least as good as z;

x y and y z x z.~ ~ ~


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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

.


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Indifference Curves Cannot

Intersect

x2

x1

xy

z

I1I2 From I1, x y. From I2, x z.

Therefore y z. But from I1

and I2we see y z, a

contradiction.


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Extreme Cases of Indifference

Curves; Perfect SubstitutesTwo goods are perfect subs t i tutes if

the consumer is willing to substitute

one good for the other at a constant

rate.


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Curves; Perfect Substitutes

x2

x18

8

15

15Slopes are constant at - 1.

I2

I1

Bundles in I2all have a totalof 15 units and are strictly

preferred to all bundles in

I1, which have a total ofonly 8 units in them.


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Curves; Perfect Complements

If a consumer always consumes

goods 1 and 2 together in fixedproportion (e.g. one-to-one), then the

commodities are perfect complements.


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x2

x1

I1

45o

5

9

5 9

Each of (5,5), (5,9)

and (9,5)contains

5 pairs so each isequally preferred.


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x2

x1

I2

I1

45o

5

9

5 9

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.


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When more of a commodity is always

preferred, the commodity is a good.

If less of a commodity is always

preferred then the commodity is a

bad.

If the consumer doesnt care about a

good, the good is a neutral good.


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Slopes of Indifference Curves

If every commodity is a good, then

indifference curves are negatively

sloped.


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Indifference Curves

x2

x1

x

x

x

x

x

x


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Indifference Curves

x2

x1

z x yx

y

z


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Indifference Curves

x2

x1

xAll bundles in I1arestrictly preferred to

all in I2.

y

z

All bundles in I2are

strictly preferred toall in I3.

I1

I2

I3


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Indifference Curves

x2

x1

Weakly preferred

set: the set of

bundles that are atleast as good as

the bundle x.

x

I(x)


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Indifference Curves

x2

x1

Str ict ly preferred

set: the set of

bundles that arestrictly preferred to

x.

x

I(x)


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Bad

Good

One good and one

bad a

positively slopedindifference curve.


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Preferences Exhibiting Satiation

A bundle strictly preferred to any

other is a sat iat ion poin t or a bl iss

po in t.

What do indifference curves look like

for preferences exhibiting satiation?


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Indifference Curves Exhibiting

Satiation

x2

x1

Satiation

(bliss)point


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Satiation

x2

x1

Be

tter

Satiation

(bliss)point


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Satiation

x2

x1

Be

tter

Satiation

(bliss)point


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Indifference Curves for Discrete

Commodities

A commodity is in f in i tely div is ib le if

it can be acquired in any quantity;

e.g.water or cheese.

A commodity is discreteif it comes

in unit lumps of 1, 2, 3, and so on;

e.g.aircraft, ships and refrigerators.


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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?


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Indifference Curves With a

Discrete Good

Gas-

oline

Aircraft0 1 2 3 4

Indifference curves

are collections ofdiscrete points.


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Well-Behaved Preferences

A preference relation is well-

behaved

if it is

monoton ic and convex.Monotonic i ty: More of any

commodity is always preferred (i.e.

no satiation and every commodity is

a good).


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Well-Behaved Preferences --

Convexity.

x2

y2

x2+y2

2

x1 y1x1+y1 2

x

y

z = x+y2

is strictly preferredto both x and y.


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Convexity.

x2

y2

x1 y1

x

y

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

for all 0 < t < 1.


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Convexity.

x2

y2

x1 y1

x

y

Preferences are strictly convex

when all mixtures z

are strictly

preferred to theircomponent

bundles x and y.

z


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Weak Convexity.

x

y

z

Preferences are

weakly convex if at

least one mixture z

is equally preferredto a component

bundle.

xz

y


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Non-Convex Preferences

x2

y2

x1 y1

zThe mixture zis less preferred

than x or y.


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More Non-Convex Preferences

x2

y2

x1 y1

zThe mixture zis less preferred

than x or y.


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The slope of an indifference curve is

known as the marg inal rate o f

subs t i tut ion (MRS).

How can a MRS be calculated?


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Marginal Rate of Substitution

x2

x1

x

MRS at x

is the

slope of the

indifference

curve at x

.

MRS is the rateat which the

consumer is

just willing to

exchangecommodity 2 for

a small amount

of commodity 1.


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MRS & Ind. Curve Properties

Good 2

Good 1

Two goods

a negatively sloped

indifference curve

MRS < 0.


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MRS & Ind. Curve Properties

Good 2

Bad 1

One good and one

bad a

positively slopedindifference curve

MRS > 0.

Di i i hi M i l R f


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Diminishing Marginal Rate of

SubstitutionGood 2

Good 1

MRS = - 5

MRS = - 0.5

Strictly convex indifference

curves: MRS decreases inabsolute value with x1.

Ch03 Consumer Preferences

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