Course: Microeconomics Text: Varian’s Intermediate

Microeconomics

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Last chapter we talk about what is affordable or feasible to consumers.

This time we talk about preferences: what the consumer like more and what they like less.

As a rational agent, a consumer chooses the option in the budget set that is highest in their preference order (i.e. one likes the most).

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Let x, y are consumption bundles. denotes strict preference so

x y means that bundle x is preferred 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 as much as 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.

It has no indication of how much they like one versus the other.

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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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Completeness: For any two bundles x and y it is always possible to make the statement that either x y or y x.

Bundles are always comparable. If both are true, then they are

indifferent 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: Ifx 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.

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: y x’}.

Weakly preferred set: bundles that are weakly preferred to x’. {y: y x’}.

~

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xx22

xx11

x”x”

x”’x”’

x’ x’ x” x” x”’ x”’x’

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 to x, and all bundles southwest of the indifference curve are less preferred to x.

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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

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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

II11

I2

From IFrom I11, x , x y. y. From IFrom I22, x , x z. z.Therefore y Therefore y z. But z. But because Ibecause I11 and I and I22 represent represent distinct level of distinct level of preference, we seepreference, 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 a good.

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.

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Better

Better

Wors

Wors

ee

Good 2Good 2

Good 1Good 1

Two goodsTwo goodsa negatively a negatively sloped sloped indifference indifference curve.curve.

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If less of a commodity is always preferred then the commodity is a bad.

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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Bette

Bette

rr

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One good and One good and oneonebad a bad a positively sloped positively sloped indifference indifference curve.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

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 don’t use and do not care about their existence.

If one commodity is neutral, the other is good, the indifference curve would be vertical / horizontal.

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If a consumer always regards units of commodities 1 and 2 as equivalent, then the commodities are perfect substitutes.

Only the total amount (or a weighted sum) of the two commodities in bundles determines their preference rank-order.

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

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xx22

xx1188

88

1515

1515Slopes are constant at - 1.Slopes are constant at - 1.

I2

I1

Bundles in IBundles in I22 all have a all have a totaltotalof 15 units and are of 15 units and are strictlystrictlypreferred to all bundles preferred to all bundles inin I I11, which have a total , which have a total ofof only 8 units in them. only 8 units in them.

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

Only the number of pairs of units of the two commodities determines the preference rank-order of bundles.

E.g.: left shoes/right shoes; computer and monitor.

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xx22

xx11

I1

4545oo

55

99

55 99

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

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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.

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A bundle strictly preferred to any others is a satiation point or a bliss 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

BetterBette

Betterr

Bett

eB

ett

err

SatiationSatiation(bliss)(bliss)pointpoint

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xx22

xx11

BettBett

erer

BettBetterer

Bett

Bett

er

er

SatiationSatiation(bliss)(bliss)pointpoint

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

A commodity is discrete if 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 an infinitely divisible good (gasoline) while commodity 1 is a discrete good (aircraft). What do indifference “curves” look like?

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GasolinGasolinee

AircraftAircraft00 11 22 3 44

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

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Typical assumptions of preferences A preference relation is “well-

behaved” if it is monotonic and convex.

Monotonicity: More of any commodity is always preferred (i.e. no satiation and every commodity is a good).

This implies a negatively sloped IC.35

Convexity: Mixtures of bundles are (at least weakly) preferred to the bundles themselves. E.g., the 50-50 mixture of the 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++yy1122

x

y

z = x+y2

Is (weakly) Is (weakly) preferred to preferred to both x and y.both 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 are strictly convex when all mixtures z are strictly preferred to their component bundles x and y.

z

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xx22

yy

22 xx11 yy11

zz

Better

The mixture zThe mixture zis less is less preferredpreferredthan x or y.than x or y.One likes One likes both, but does both, but does want to want to consume consume together.together.

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xx22

yy

22 xx11 yy11

zz

BetterThe mixture zThe mixture zis less preferredis less preferredthan 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.

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The negative of the slope of an indifference curve is its marginal rate of substitution (MRS).

Note: this is slightly different from the textbook, but my definition is more popular.

This represents the maximum amount of x2 one is willing to give up per unit of x1 at a specific consumption bundle.

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xx22

xx11

x’x’

MRS at x’ is the MRS at x’ is the (negative) slope of (negative) slope of thetheindifference curve at indifference curve at x’x’

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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’

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xx22

x1

dxdx22

dxdx11

-dx-dx22 = MRS = MRS dx dx11 so, so, at x’, MRS is the at x’, MRS is the raterate at which the at which the consumer is only just consumer is only just willing to exchange willing to exchange commodity 2 for a commodity 2 for a small amount of small amount of commodity 1.commodity 1.

x’x’

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Better

Better

Wors

Wors

ee

Good 2Good 2

Good 1Good 1

Two goodsTwo goodsa negatively a negatively sloped sloped indifference curveindifference curveMRS > 0.MRS > 0.

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Bette

Bette

rr

Wors

e

Wors

e

Good 2Good 2

Bad 1Bad 1

One good and One good and oneonebad a bad a positively sloped positively sloped indifference curveindifference curveMRS < 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

Good 2Good 2

Good 1Good 1

MRS = 5MRS = 5

MRS = 0.5MRS = 0.5

MRS always MRS always decreases with xdecreases with x11 if if and only if and only if preferences are preferences are strictlystrictly convexconvex..

We call it a diminishing marginal rate of substitution. 49

xx11

xx22 MRS = 0.5

MRS = 5

MRS increasesMRS increasesas xas x11 increases increasesnon-convex non-convex preferencespreferences

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xx22

xx11

MRS= 0.5 MRS = 2

MRS is not always MRS is not always decreasing as xdecreasing as x11 increases non-increases non-convex preferences.convex preferences.MRS=1.

5

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In this chapter, we talk about how we can specify consumer’s 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.

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We have talked about preference and indifference curve in this chapter.

To put preference in a more mathematically convenient way, we introduce the utility function in the coming chapter.

Then we can put together preference/utility and budget constraint to analyze consumer choices.

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