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Course: MicroeconomicsText: VariansIntermediate

Microeconomics

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Last chapter we talk about what isaffordable or feasible to consumers.

This time we talk aboutpreferences:what the consumer like more and whatthey like less.

As a rational agent, a consumer chooses

the option in the budget set that ishighestin their preference order (i.e.one likes the most).

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Let x, y are consumption bundles. denotesstrict preferenceso

x y means that bundle x is preferred

strictly to bundle y.denotesindifference; xy means x

and y are equally preferred.

denotesweak preference;x y means x is preferred at least as

much as is y.

~

~

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Strict preference, weak preference and

indifference are all preference relations.

Particularly, they areordinalrelations;i.e.they

state only theorderin which bundles are

preferred.

It has no indication of how much they like one

versus the other.

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x y and y x imply xy.

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

~ ~

~ ~

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Completeness: For any two bundles xand y it is always possible to make thestatement that either

x yor y x.

Bundles are always comparable. If both are true, then they areindifferent to the individual.

~

~

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Reflexivity: Any bundle x is always at

least as preferred as itself;i.e.

x x.~

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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.

~ ~ ~

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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: yx}.

Weakly preferred set: bundles that are

weakly preferred to x. {y: y x}.~

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xx22

xx11

xx

xx

xx xx xxx

The consumer is

indifferent between

every point on the

indifference curve.

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xx22

xx11

zz xx yy

x

y

z

If consumer prefers more

to less for each goods, all

bundles on the northeast

of the indifference curve

are strictly preferred tox, and all bundles

southwest of the

indifference curve are

less preferred to x.

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x2

x1

x

All bundles in I1arestrictly preferred to

all in I2.

y

z

All bundles in I2

are strictlypreferred to all in

I3.

I1

I2

I3

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x2

x1

I(x)

x

I(x)

WP(x), the set of

bundles weakly

preferred to x.

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x2

x1

WP(x), the set of

bundles weakly

preferred to x.WP(x)

includes

I(x).

x

I(x)

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x2

x1

SP(x), the set of

bundles strictly

preferred to x, does not

include

I(x).

x

I(x)

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xx22

xx11

xxyy

zz

II11I2

From IFrom I11, x, x y.y.From IFrom I22, x, x z.z.

Therefore yTherefore y z. But becausez. But because

II11and Iand I22represent distinctrepresent distinct

level of preference, we seelevel of preference, we see

y z, a contradiction.y z, a contradiction.

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When more of a commodity is always preferred,

the commodity is agood.

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

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Better

Better

W

orse

Worse

Good 2Good 2

Good 1Good 1

Two goodsTwo goods

a negatively slopeda negatively sloped

indifference curve.indifference curve.

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If less of a commodity is always preferred then

the commodity is abad.

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.

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Bett

er

Better

Worse

Worse

Good 2Good 2

Bad 1Bad 1

One good and oneOne good and onebad abad a

positively slopedpositively sloped

indifference curve.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

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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 dont 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

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If a consumer always regards units ofcommodities 1 and 2 as equivalent, thenthe commodities areperfect substitutes.

Only thetotal amount (or a weighted sum)of the two commodities in bundlesdetermines their preference rank-order.

Example: orange juice of two differentbrands. Apartment in different locations.

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xx22

xx1188

88

1515

1515

Slopes are constant at - 1.Slopes are constant at - 1.

I2

I1

Bundles in IBundles in I22all have a totalall 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.

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If a consumer always consumescommodities 1 and 2 in fixed proportion(e.g. one-to-one), then the commodities

areperfect complements. Only thenumber of pairsof units of thetwo commodities determines the

preference rank-order of bundles. E.g.: left shoes/right shoes; computerand monitor.

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xx22

xx11

I

1

4545oo

55

99

55 99

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

and(9,5)contains

5 pairs so each is

equally preferred.

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xx22

xx11

I2

I

1

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 9pairs.

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A bundle strictly preferred to any others is a

satiation pointor abliss point.

The satiation point is the best bundle. More of

anything are not better.

What do indifference curves look like for

preferences exhibiting satiation?

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xx22

xx11

SatiationSatiation

(bliss)(bliss)pointpoint

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xx22

xx11

Better

BetterBetter

Better

Be

tter

Be

tter

SatiationSatiation

(bliss)(bliss)pointpoint

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xx22

xx11

Bette

Bette

rr

Bette

Bette

rr

Be

tte

Bette

rr

SatiationSatiation

(bliss)(bliss)pointpoint

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A commodity isinfinitely divisibleif it

can be acquired in any quantity;e.g.

water or cheese.

A commodity isdiscreteif it comes in

unit lumps of 1, 2, 3, and so on;e.g.

aircraft, ships and refrigerators.

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Suppose commodity 2 is aninfinitely

divisiblegood (gasoline) while

commodity 1 is adiscretegood (aircraft).What do indifference curves look like?

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GasolineGasoline

AircraftAircraft00 11 22 3 44

Indifference curvesIndifference curves

are collections ofare collections ofdiscrete points.discrete points.

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Typical assumptions of preferencesA preference relation is well-behaved if

it is monotonicandconvex.

Monotonicity: More of any commodity is

always preferred (i.e.no satiation andevery commodity is a good).

This implies a negatively sloped IC.35

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Convexity: Mixtures of bundles are (at

least weakly) preferred to the bundles

themselves. E.g., the 50-50 mixture ofthe bundles x and y is

z = (0.5)x + (0.5)y.

z is at least as preferred as x or y.

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xx22

yy22

xx22+y+y22

22

xx11 yy11xx11+y+y

11

22

x

y

z =x+y2

Is (weakly)Is (weakly)preferred to bothpreferred to both

x and y.x and y.

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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.

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xx22

yy22

xx11 yy11

x

y

Preferences arestrictly convex

whenallmixtures z

arestrictly

preferredto their component

bundles x and y.

z

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xx22

yy22

xx11 yy11

zz

Better

The mixture zThe mixture z

is less preferredis less preferred

than x or y.than x or y.One likes both,One likes both,

but does wantbut does want

to consumeto consume

together.together.

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xx22

yy22

xx11 yy11

zz

Better

The mixture zThe mixture zis less preferredis less preferred

than x or y.than x or y.

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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

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The negative of the slope of an indifference

curve is itsmarginal rate of substitution(MRS).

Note: this is slightly different from the textbook, but my

definition is more popular.

This represents the maximum amount of x2oneis willing togive upper unit of x1at a specific

consumption bundle.

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xx22

xx11

xx

MRS at x is theMRS at x is the

(negative) slope of the(negative) slope of the

indifference curve at xindifference curve at x

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xx22

xx11

MRS at x isMRS at x is

lim -{lim -{xx22//xx11}}

xx11 0 0

= -dx= -dx22/dx/dx11at xat xxx22

xx11

xx

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xx22

x1

dxdx22dxdx11

-dx-dx22

= MRS= MRS dxdx11

so, atso, at

x, MRS is thex, MRS is theraterateatat

which the consumer iswhich the consumer is

only just willing toonly just willing to

exchange commodity 2exchange commodity 2for a small amount offor a small amount of

commodity 1.commodity 1.

xx

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Better

Be

tter

Worse

Worse

Good 2Good 2

Good 1Good 1

Two goodsTwo goods

a negatively slopeda negatively sloped

indifference curveindifference curve

MRS > 0.MRS > 0.

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Better

Better

Worse

Worse

Good 2Good 2

Bad 1Bad 1

One good and oneOne good and one

bad abad a

positively slopedpositively slopedindifference curveindifference curve

MRS < 0.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

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Good 2Good 2

Good 1Good 1

MRS = 5MRS = 5

MRS = 0.5MRS = 0.5

MRS always decreasesMRS always decreases

with xwith x11if and only ifif and only if

preferences are strictlypreferences are strictly convexconvex..

We call it adiminishing marginal rate

of substitution. 49

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xx11

xx22 MRS = 0.5

MRS = 5

MRS increasesMRS increases

as xas x11increasesincreases

non-convex preferencesnon-convex preferences

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xx22

xx11

MRS

= 0.5MRS = 2

MRS is not alwaysMRS is not always

decreasing as xdecreasing as x11increasesincreases

non-convexnon-convex

preferences.preferences.MRS=1.5

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In this chapter, we talk about how we canspecify consumers 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

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We have talked about preference andindifference curve in this chapter.

To put preference in a more mathematicallyconvenient way, we introduce the utility

function in the coming chapter.

Then we can put together preference/utility and

budget constraint to analyze consumer choices.