Chapter 3 Preferences

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Chapter 3 Preferences

• Course Microeconomics
• Text Varians Intermediate Microeconomics

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Introduction

• 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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Preference Relations

• 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 preferencex y means x is
  preferred at least as much as is y.

p
p
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Preference Relations

• 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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Preference Relations

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

p
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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 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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Assumptions about Preference Relations

• Reflexivity Any bundle x is always at least as
  preferred as itself i.e.
  x x.

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Assumptions about Preference Relations

• 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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Indifference Curves

• 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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Indifference Curves
x2
x x x
x
The consumer is indifferent between every point
on the indifference curve.
x
x
x1
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Indifference Curves
z x y
p
p
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.
x2
x
z
y
x1
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Indifference Curves
I1
All bundles in I1 are strictly preferred to all
in I2.
x2
x
z
I2
All bundles in I2 are strictly preferred to all
in I3.
y
I3
x1
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Indifference Curves
x2
WP(x), the set of bundles weakly preferred to
x.
x
I(x)
I(x)
x1
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Indifference Curves
x2
WP(x), the set of bundles weakly preferred to
x.
x
WP(x) includes I(x).
I(x)
x1
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Indifference Curves
x2
SP(x), the set of bundles strictly preferred
to x, does not include
I(x).
x
I(x)
x1
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Indifference Curves Cannot Intersect
From I1, x y. From I2, x z. Therefore y
z. But because I1 and I2 represent distinct
level of preference, we see y z, a
contradiction.
I2
x2
I1
p
x
y
z
x1
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Goods

• 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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Slopes of Indifference Curves
Good 2
Two goodsa negatively sloped indifference curve.
Better
Worse
Good 1
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Bads

• 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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Slopes of Indifference Curves
Good 2
One good and onebad a positively
sloped indifference curve.
Better
Worse
Bad 1
If you want more of the good, you also have to
get more of the bad so that you are indifferent
between them.
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Neutrals

• 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.

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Slope of Indifference 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 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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Extreme Cases of Indifference Curves Perfect
Substitutes
x2
Slopes are constant at - 1.
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I2
Bundles in I2 all have a totalof 15 units and
are strictly preferred to all bundles in
I1, which have a total of only 8 units
in them.
8
I1
x1
8
15
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Extreme Cases of Indifference Curves Perfect
Complements

• 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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Extreme Cases of Indifference Curves Perfect
Complements
x2
Each of (5,5), (5,9) and (9,5) contains5 pairs
so each is equally preferred.
45o
9
5
I1
x1
5
9
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Extreme Cases of Indifference Curves Perfect
Complements
x2
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.
45o
9
I2
5
I1
x1
5
9
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Preferences Exhibiting Satiation

• 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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Indifference Curves Exhibiting Satiation
x2
Satiation(bliss)point
x1
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Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
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Indifference Curves Exhibiting Satiation
x2
Better
Better
Satiation(bliss)point
Better
x1
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Indifference Curves for Discrete Commodities

• 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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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
Gasoline
Indifference curvesare collections ofdiscrete
points.
Aircraft
0
1
2
3
4
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Well-Behaved Preferences

• 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.

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Well-Behaved Preferences

• 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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Well-Behaved Preferences -- Convexity.
x
x2
xy
Is (weakly) preferred to both x and y.
x2y2
z
2
2
y
y2
x1y1
x1
y1
2
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Well-Behaved Preferences -- Convexity.
x
x2
z (tx1(1-t)y1, tx2(1-t)y2)
is preferred to x and y for all 0 lt t lt 1.
y
y2
x1
y1
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Well-Behaved Preferences -- Convexity.
Preferences are strictly convex
when all mixtures z are
strictly preferred to their
component
bundles x and y.
x
x2
z
y
y2
x1
y1
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Non-Convex Preferences
The mixture zis less preferred than x or y. One
likes both, but does want to consume together.
x2
Better
z
y2
x1
y1
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More Non-Convex Preferences
x2
Better
The mixture zis less preferred than x or y.
z
y2
x1
y1
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Why convexity?

• 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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Slopes of Indifference Curves

• 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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Marginal Rate of Substitution
x2
MRS at x is the (negative) slope of
theindifference curve at x
x
x1
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Marginal Rate of Substitution
x2
MRS at x is lim -Dx2/Dx1 Dx1
0 -dx2/dx1 at x
x
Dx2
Dx1
x1
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Marginal Rate of Substitution
-dx2 MRS dx1 so, at x, MRS is the rate at
which the consumer is only just willing to
exchange commodity 2 for a small amount of
commodity 1.
x2
x
dx2
dx1
x1
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MRS Ind. Curve Properties
Good 2
Two goodsa negatively sloped indifference curve
Better
MRS gt 0.
Worse
Good 1
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MRS Ind. Curve Properties
Good 2
One good and onebad a positively
sloped indifference curve
Better
MRS lt 0.
Worse
Bad 1
Because instead of giving up, you have to obtain
more good 2 for you to be willing to accept more
good 1.
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MRS Ind. Curve Properties
Good 2
MRS 5
MRS always decreases with x1 if and only if
preferences are strictly convex.
MRS 0.5
Good 1
We call it a diminishing marginal rate of
substitution.
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MRS Ind. Curve Properties
x2
MRS increasesas x1 increasesnon-convex
preferences
MRS 0.5
MRS 5
x1
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MRS Ind. Curve Properties
MRS is not always decreasing as x1 increases
non-convex preferences.
x2
MRS1.5
MRS 0.5
MRS 2
x1
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Summary

• In this chapter, we talk about how we can specify
  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.

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Whats next?

• 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.