l4 preferences.ppt

8/9/2019 L4 Preferences.ppt

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Modeling Consumer Preferences


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Preferences in Economics

! We are generally interested inunderstanding :

– how choices are made and – how they can be influenced


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Preferences in Economics

! We are generally interested inunderstanding :

– how choices are made and – how they can be influenced

! Just figured out: constraints on choices! To model choice: also need to model

decision makers’

preferences .


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Preference Relations! Comparing two different consumption

bundles, x and y: – strict preference : x is more preferred

than y. – weak preference : x is as at least as

preferred as y.

– indifference : x is exactly as preferredas y.


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Important note:

Preference relations are ordinal relations;

i.e. they state only the order in whichbundles are preferred.


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Preference notation:

! denotes strict preference;x y means bundle x is preferred

strictly to bundle y.


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strictly to bundle y.! denotes indifference; x y means x

and y are equally preferred.


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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 asmuch as is y. ~~


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Some logical rules:

! x y and y x imply x y.~ ~


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Some logical rules:

! 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 : For any two bundles x

and y it is always possible to make the

statement that eitherx yor

y x.

~#

~#


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Relations! Reflexivity : Any bundle x is always at

least as preferred as itself; i.e.

x x.~#


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Relations! Transitivity : If

x is at least as preferred as y, and

y is at least as preferred as z, thenx is at least as preferred as z; i.e.

x y and y z x z.

~#

~#

~#


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Useful graphical tool:

Indifference Curves! Take a bundle x. The indifference curve

containing x is a set of all bundles y

such that y " x

! Since an indifference“

curve”

is not

always a curve a better name might bean indifference“

set”

.


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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 I 1 arestrictly preferred toall in I 2.

y

z

All bundles in I 2 are

strictly preferred toall in I 3.

I1

I2

I3


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Indifference Curves

x2

x1

the set ofbundles weakly

preferred to x(includes I(x))

x

I(x)


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Indifference Curves

x2

x1

the set ofbundles strictly

preferred to x(does not

includeI(x))

x

I(x)


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Lemma: Indifference CurvesCannot Intersect

x2

x1

xy

z

I1I2 Proof:

From I 1, x " y.

From I 2, x " z.By transitivity, y " z.


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Slopes of Indifference Curves

! When more of a commodity is alwayspreferred, the commodity is a good .

! If every commodity is a good thenindifference curves are negatively sloped .


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Slopes of Indifference CurvesGood 2

Good 1

Two goodsa negatively slopedindifference curve.


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Slopes of Indifference Curves

! If less of a commodity is alwayspreferred then the commodity is a bad .


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Slopes of Indifference CurvesGood 2

Bad 1

One good and onebad apositively slopedindifference curve.


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Extreme Cases of Indifference

Curves: Perfect Substitutes! If a consumer always regards units of

commodities 1 and 2 as equivalent,

then they are perfect substitutes

! only the total amount of the twodetermines their preference rank-order


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Curves: Perfect Substitutesx2

x18

8

15

15Slopes are constant, here -1

I2

I1

Bundles in I 2 all have a totalof 15 units and are strictlypreferred 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

commodities 1 and 2 in fixed proportion,

say, one-to-one, then the commoditiesare perfect complements

! only the number of pairs of units(or triplets, etc.) determines thepreference rank-order of bundles.


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Curves: Perfect Complementsx2

x1

I1

45 o

5

9

5 9

Each of (5,5), (5,9)and (9,5) contains

5 pairs so each isequally preferred.


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Curves: Perfect Complementsx2

x1

I2

I1

45 o

5

9

5 9

Since each of (5,5),(5,9) and (9,5)

contains 5 pairs,each is lesspreferred than thebundle (9,9) which

contains 9 pairs.


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Indifference Curves Exhibiting

Satiationx2

x1

Satiation(bliss)point


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Satiationx2

x1

B e

t t e r



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Satiationx2

x1

B e

t t e r



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Indifference Curves for Discrete

Commodities! A commodity is discrete if it comes in

unit lumps of 0,1,2,3,…,856,… and so

on (i.e. in integer amounts)

! Examples: aircraft, chairs, refrigerators


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Indifference Curves With a

Discrete Good! Suppose commodity 2 is an infinitely

divisible good (jet fuel) while commodity

1 is a discrete good (aircraft)


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Indifference Curves With a

Discrete GoodFuel

Aircraft0 1 2 3 4

Indifference“

curves”

are collections ofdiscrete points.


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Assume Well-Behaved

Preferences! We may want to simplify further and rule

out some extreme examples by

additional assumptions

! A preference relation is“

well-behaved”

if it is – monotone and convex


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Well-Behaved Preferences

! Monotonicity : More of any commodityis always preferred

(i.e. no satiation and every commodity is agood).


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Well-Behaved Preferences

! Convexity : Mixtures of bundles are (atleast weakly) preferred to the bundles

themselves.

! E.g., the 50-50 mixture of the bundles xand y is

z = (0.5)x + (0.5)y.z is at least as preferred as x or y.


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Well-Behaved Preferences -

Convexity

x2

y2

x2+y 22

x1 y1x1+y 12

x

y

z = x+y2

z is strictlypreferred to both xand y.


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Convexity

x2

y2x1 y1

x

y

z =(tx 1+(1-t)y 1, tx 2+(1-t)y 2)is preferred to x and yfor all 0 < t < 1.


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Weak Convexityx

y

z

Preferences areweakly convex if atleast one mixture z isequally preferred to acomponent bundle.x z

y


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Non-Convex Preferences

x2

y2

x1 y1

zThe mixture zis less preferredthan x or y.


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Marginal Rate of Substitution


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! The slope of an indifference curve is itsmarginal rate of substitution

(MRS)


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x2

x1

x

MRS at x

is the slope of theindifference curve at x


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x2

x1

MRS at x’:

lim { # x2 /# x1}# x1 0

# x2

# x1

x


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x2

x1

MRS at x’:

lim { # x2 /# x1} = dx 2 /dx 1 at x

# x1 0# x2

# x1

x


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x2

x1

dx 2dx 1

dx 2 = MRS $ dx 1 at x’

x


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x2

x1

dx 2dx 1

dx 2 = MRS $ dx 1 at x’

so MRS is the rate at which

the consumer is only justwilling to exchangecommodity 2 for a smallamount of commodity 1 at x’

x


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MRS & Ind. Curve PropertiesGood 2

Good 1

Two goodsa negatively slopedindifference curve

MRS < 0.


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Bad 1

One good and onebad apositively slopedindifference curve

MRS > 0.


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Good 1

MRS = - 5

MRS = - 0.5

MRS always increases with x 1 (becomes less negative) if andonly if preferences are strictly

convex


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MRS & Ind. Curve Properties

x2

x1

MRS= - 0.5

MRS = - 1

MRS = - 2

If MRS does not always increase (say, with x 1)! > nonconvex preferences (in x

1)

l4 preferences.ppt

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