economics_102_lecture_3_preferences_rev.pdf

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Lecture 3 - Theory of the

Consumer; Preferences

Consumer preferences: Notations and preference

relations

Assumptions on preference relations

Indifference curves as a graphical representationof consumer preferences.

Assumptions for well behaved indifference curves

Marginal Rate of Substitution


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Consumption bundles

Give a complete list of goods and services that areinvolved in the consumer choice problem.

In other words, consumers pick from amongconsumption bundles.

While we assume that the consumption bundle iscomplete, we usually represent the bundle interms of two goods.

We can represent consumption bundles as (x1,x2)or byX.

We assume that consumers can rank

consumption bundles as to their

desirability.

consumers prefer one good over the other in

the sense that they would choose one bundleover another given the opportunity

the idea of preference is based on the idea of

behavior


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We use the following symbols to denotethese rankings:

Strict preference

The consumer definitely wants theX bundlerather than the Y bundle.

The consumer always choosesXover YwhenXis available.

),(to),(prefersstrictlyConsumer

),(),(

preferenceStrictly

2121

2121

yyxx

yyxx

Indifference

Indifference means that the consumer would be

equally satisfied consuming bundleXor bundle

Y.

When bothXand Yare available, consumer

would choose either bundle

),(and),(betweentindifferenisConsumer

),(~),(

~ceIndifferen

2121

2121

yyxx

yyxx


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Weakly prefers

Consumer prefers or is indifferent between

),(),(

preferenceWeak

2121 yyxx

2121 ,and, yyxx

Relations between these rankings:

If a consumer weakly prefersXto Yand weaklyprefers YtoX, then the consumer is indifferentbetweenXand Y.

If a consumer thinks thatXis at least as good as Yand Yis at least as good asX, then the consumermust be indifferent between the two bundles.

),(~),(then),,(),(and),(),(If 212121212121 yyxxxxyyyyxx


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If the consumer thinks thatXis at least

better than Ybut is not indifferent between

Xand Y, then the consumer must strictly

preferXto Y.

),(),(then),,(~),(notand),(),(If 212121212121 yyxxxxyyyyxx

Complete

any two bundles can be compared, and

consumers can make a choice between any

two bundles

Reflexive: Any bundle is at least as good

as itself

bothor),(),(or),(),(Either 21212121 xxyyyyxx

),(),( 2121 xxxx


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Transitive

If X is as good as Y, and Y is as good as Z, then X is

at least as good as Z

),(),(then),,(),(and),(),(If 212121212121 zzxxzzyyyyxx

We represent preferences graphically by

indifference curves or indifference sets

Indifference curve/set through a bundle

consists of all consumption bundles that leave

the consumer indifferent to a given bundle Fig 1

i.e., the set of all bundles equally preferred to a

reference bundle x is the indifference curve

containing x; the set of all bundles y ~ x.

Weakly preferred set- all consumption bundles

that are weakly preferred to a specific bundle


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x2

x1

x

x

x ~ x ~ xx

x2

x1

z x yp

x

y

z


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x2

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

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)

x2

x1

SP(x), the set of

bundles strictly

preferred to x,

does not

includeI(x).

x

I(x)


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1

Indifference curves representing distinct levels of

preference do not cross.

Intersecting indifference curves violate the transitivity

assumption about preferences.

SupposeXis preferred to Ybut indifferent toZ, Yisindifferent toZand the axiom of transitivity implies that

Xis indifferent to Ywhich is not the case

x2

x1

x y

z

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

Therefore y ~ z.


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x2

x1

xy

z

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

Therefore y ~ z. But from I1and I2 we see y z, a

contradiction.

p

Since indifference curves are a way to describe

preferences, then reasonable preferences can be

described by indifference curves.

To construct indifference curves ask - for a given

change in good 1, how does the consumption of good2 have to change so that the consumer is indifferent

between the old consumption level and the new

consumption level?


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1

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.

Good 2

Good 1

Two goods

a negatively sloped

indifference curve.


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1

Bads

A bad is a commodity that the consumer doesnt like

Assume that there is some tradeoff between the twogoods

Suppose a bundleXwithx1 as the good andx2 as thebad. To keep a consumer on the same indifferencecurve, we have to give him more of good 1 in order tocompensate him for having to put up with the bad

Bads are represented by upward sloping indifference

curves The direction of increasing preference is down and to

the right, toward the direction of increasedconsumption of the good and toward decreasedconsumption of the bad

Good 2

Bad 1

One good and one

bad a

positively sloped

indifference curve.


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1

Perfect substitutes

If a consumer is willing to substitute one good for

another at a constant rate, then we consider the two

goods as perfect substitutes.

For perfect substitutes, indifference curves have a

constant slope.

Since only the total quantities matter, the direction of

increasing preference is to the right.

x2

x18

8

15

15Slopes are constant at - 1.

I2

I1

Bundles in I2 all have a total

of 15 units and are strictly

preferred to all bundles inI1, which have a total of

only 8 units in them.


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1

Perfect complements

Goods that are consumed together in fixed proportions.

The indifference curves for perfect complements are L-shaped, with the vertex occurring where the ratio of theamount of good 1 to the amount of good 2 are at thefixed proportion

Direction of increasing preferences is up and to theright. Increasing both the quantities of both goods will

move the consumer to a more preferred condition.

x2

x1

I1

45o

5

9

5 9

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

and (9,5) contains

5 pairs so each is

equally preferred.


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1

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.

Neutral

Neutral goods are goods that the consumer

dont care about.

the indifference curves are vertical lines


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1

x2

x1

Satiation point or bliss point most preferred bundle ofgoods

Represented by

The closer the consumer is to that bundle, the better offhe is in terms of preferences.

Points nearer the bliss points lie on higher indifferencecurves while those that are farther lie on lowerindifference curves

Indifference curve has a negative slope when he has toolittle or too much of both goods.

Indifference curve has a positive slope when he has toomuch of one of the goods.

Direction of preference is toward the bliss point.

),( 21 xx


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1

x2

x1

Satiation

(bliss)

point

x2

x1

Better

Satiation

(bliss)

point


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1

x2

x1

B

etter

Satiation

(bliss)

point

We will limit our analysis to that region

where we have less of what we want. Most

likely, people are below rather than above

their bliss points.


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2

Discrete goods consumption is usually in integerunits

preferences can still be represented as continuousvariables by a suitable redefinition of the good inquestion, e.g., number of automobiles (discrete) vs.hours spent in a car (continuous).

If not then preferences can still be represented byindifference curves and weakly preferred set Points with the same color are bundles that are indifferent

to one another

Vertical lines represent bundles that are at least as good

Continuou

s good

Discrete good

0 1 2 3 4

Indifference curves

are collections of

discrete points.


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2

Monotonicity

This assumption just states that more is

better

IfXis a bundle of goods and Yis a bundle of

goods with at least as much as both goods

and more of one, then Yis preferred toX.

XY

xyxyyyxx

:then

,and,such that),(),,(bundlestwoSuppose 22112121

Monotonicity implies that indifference curveshave a negative slope.

From a bundleX, moving up and to the right impliesmoving to a more preferred position

Moving down and to the left implies moving to a less

preferred position If we are moving to an indifferent position, then we

must be moving rightward and down and or movingleftward and up.


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2

x2

x1

x

Better bundles

Worse bundles

Convexity

Assumes that averages are preferred to extremes

The distance fromXto the average bundle is just afraction tof the distance from theXbundle to the Y

bundle along the line that connects the two bundles.

),())1(,)1((:Then

),(~),(Assume

10weight,abeLet t

212211

2121

xxyttxyttx

yyxx

t


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2

Geometrically, it means that the bundlesweakly preferred to X is a convex set.

A convex set has the property that if you takeany two points in the set and draw a linesegment connecting these two points, then theline segment is entirely in the set.

Reason for assuming convex preferences-most goods are consumed together Example of nonconvex preferences- two goods that

somebody might not prefer to consume together.

Strictly convex - Assumes that theweighted average of two bundles isstrictly preferred to the two extremebundles. convex preferences- indifference curves

may have flat spots strictly convex preferences have rounded

indifference curves

indifference curves for perfect substitutesare convex but not strictly convex.


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2

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.

x2

y2

x2+y2

2

x1 y1x1+y1

2

x

y

z =x+y

2

is strictly preferred

to both x and y ift=.50


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2

x2

y2

x1 y1

x

y

Preferences are strictly convex

when all mixtures z

are strictly

preferred to their

component

bundles x and y.

z

x

y

z

Preferences are

weakly convex ifat

least one mixture z

is equally preferred

to a componentbundle.

xz y


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2

x2

y2

x1 y1

z

The mixture z

is less preferred

than x or y.

x2

y2

x1 y1

z

The mixture z

is less preferred

than x or y.


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2

Marginal rate of substitution slope

of the indifference curve. The slope

denotes the rate at which the

consumer is just willing to substitute

one good for another.

Suppose we take a little bit of good 1, x1 from theconsumer and give him x2, an amount that is justsufficient to put him back on his indifference curve

The ratio x2/x1 is the rate at which the consumeris willing substitute good 2 for good 1.

If we think of the changes as very small changes, thenthe ratio measures the marginal rate of substitutionof good 2 for good 1. As x1 gets smaller, the ratioapproaches the slope of the indifference curve


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2

x2

x1

MRS at x is

lim {x2/x1}x1 0

= dx2/dx1 at xx2

x1

x

x2

x1

dx2dx1

dx2 = MRS dx1 so, at x, MRS isthe rate at which the consumer is

only just willing to exchange

commodity 2 for a small amount of

commodity 1.x


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2

x2

x1

x

MRS at x is the slope of the

indifference curve at x

MRS is typically a negative number. Monotonicpreferences imply that indifference curves have anegative slope.

Marginal rate of substitution measures the rate atwhich the consumer is just on the margin of tradingor not trading. At any rate of exchange other thanthe MRS, the consumer would want to trade onegood for another. If the rate of exchange equals the

MRS, then the consumer stays put.


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3

x2

x1

x

E

Other interpretation of the MRS: Slope of the

indifference curve can also be interpreted as the

marginal willingness to pay

Using the concept of a composite good or spending

on all other goods - interpretation is how many

pesos you are willing to give up spending on other

goods in order to increase consumption of one good.

marginal willingness to pay is different from what

you have to pay


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3

Behavior of the MRS

MRS for different preferences

perfect substitutes = constant MRS

perfect complements = MRS is zero or infinity

neutrals = MRS is everywhere infinite, youcannot compensate one for consuming anextra good because he doesnt care about theother good

Since monotonicity of preferences implies thatslope of indifference curves are negative - thenMRS always involves reducing the consumptionof one good in order to get more of another

Convexity, in particular strict convexity, implies

that the slope of indifference curves decreases in

absolute value as we increase x1.

Diminishing marginal rates of substitution rate at

which a person is willing to trade good 1 for good 2

decreases as we increase the amount of good 1


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Good 2

Good 1

MRS = - 5

MRS = - 0.5

MRS always increases with x1 (becomes less

negative) if and only if preferences are strictly

convex.

economics_102_lecture_3_preferences_rev.pdf

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