Chapter 3
Preferences
Introduction




The economic model of consumer behavior is very
simple: people choose the best things they can afford.
The last chapter was devoted to clarifying the meaning
of “can afford”.
This chapter will be devoted to clarifying the economic
concept of “best things”. A consumer always chooses
his most preferred one from his set of available
alternatives.
So to model consumers’ choices we must model their
preferences.
2
Preference Relations
Let x, y are consumption bundles.

denotes strict preference: x y means that
bundle x is strictly preferred to bundle y.
 ~ denotes indifference: x ~ y means x and y are
equally preferred.
 f denotes weak preference: x f y means x is
~
~
preferred at least as much as is y.

p
3
p
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.
 x f y and y f x imply x ~ y.
~
~
 x f y and (not y f x) imply x
y.

~
p
~
4
Assumptions about Preference
Relations
Completeness: For any two bundles x and y
it is always possible to make the statement
that either
x f y
~
or
y f x.
~
 Bundles are always comparable.
 Again, if both are true, then they are
indifferent to the individual.

5
Assumptions about Preference
Relations

Reflexivity: Any bundle x is always at least as
preferred as itself; i.e.
x
x.
f
~
6
Assumptions about Preference
Relations

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 f z
f
~
~
x f z.
~
7
Indifference Curves
Take a reference bundle x’. The set of all
bundles equally preferred to x’ is the
indifference curve (set) containing x’; i.e., the
set of all bundles {y: y ~ x’}.
 Weakly preferred set: bundles that are weakly
preferred to x’. {y: y fx’}.

~
8
Indifference Curves
x2
x’
x’ ~ x” ~ x”’
x”
x”’
x1
9
Indifference Curves
x
z
y
p
z
x
p
x2
y
If the consumer prefers
more to less for each good,
all bundles to the northeast
of the indifference curve are
strictly preferred to x, and
all bundles to the southwest
of the indifference curve are
less preferred to x.
x1
10
Indifference Curves
I1
x2
x
z
I2
y
I3
All bundles in I1 are
strictly preferred to all
in I2.
All bundles in I2 are
strictly preferred to
all in I3.
x1
11
Indifference Curves
x2
x
WP(x), the set of
bundles weakly
preferred to x.
I(x)
I(x’)
x1
12
Indifference Curves
x2
x
WP(x), the set of
bundles weakly
preferred to x.
WP(x)
includes
I(x)
I(x).
x1
13
Indifference Curves
x2
x
SP(x), the set of
bundles strictly
preferred to x,
does not
include
I(x)
I(x).
x1
14
Indifference Curves Cannot
Intersect
x2
I1
I2 From I1, x ~ y.
From I2, x ~ z.
Therefore y ~ z. However, I1 and I2
represent different levels of preference.
Contradiction!
x
y
z
x1
15
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.
16
Slopes of Indifference Curves
Good 2
Two goods
a negatively sloped
indifference curve.
Good 1
17
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.

18
Slopes of Indifference Curves
Good 2
One good and one
bad
a positively sloped
indifference curve.
Bad 1
19
Neutrals
If one just do not care about whether or how
much to have a commodity, then the
commodity 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.

20
Slopes of Indifference Curves
21
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.
 e.g. orange juice of two different brands.

22
Perfect Substitutes
x2
15
I2
8
I1
Slopes are constant at - 1.
Bundles in I2 all have a total
of 15 units and are strictly
preferred to all bundles in
I1, which have a total of
only 8 units in them.
x1
8
15
23
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 rankorder of bundles.
 e.g. left shoes/right shoes.

24
Perfect Complements
x2
45o
9
5
Each of (5,5), (5,9) and
(9,5) contains
5 pairs so each is
equally preferred.
I1
5
9
x1
25
Perfect Complements
x2
45o
9
5
I1
5
9
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs, each
is less preferred than
I2 the bundle (9,9) which
contains 9 pairs.
x1
26
Preferences Exhibiting Satiation

A bundle strictly preferred to any other is a
satiation point or a bliss point.

What do indifference curves look like for
preferences exhibiting satiation?
27
Indifference Curves Exhibiting
Satiation
x2
Satiation
(bliss)
point
x1
28
Indifference Curves Exhibiting
Satiation
x2
Better
Satiation
(bliss)
point
x1
29
Indifference Curves Exhibiting
Satiation
x2
Better
Satiation
(bliss)
point
x1
30
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.

31
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?
32
Indifference Curves With a Discrete Good
Gasoline
Indifference “curves”
are collections of
discrete points.
0
1
2
3
4 Aircraft
33
Well-Behaved 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).
 Monotonicity implies that indifference curves
are negatively sloped.

34
Well-Behaved Preferences

Convexity: Mixtures of bundles are (at least
weakly) preferred to the bundles themselves. For
example, 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.
35
Well-Behaved Preferences -Convexity
x
x2
x+y
z=
2
x2+y2
2
z is preferred to both x
and y.
y
y2
x1
x1+y1
2
y1
36
Well-Behaved Preferences -Convexity
x
x2
z =(tx1+(1-t)y1, tx2+(1-t)y2)
is preferred to x and y for all
0 < t < 1.
y
y2
x1
y1
37
Well-Behaved Preferences -Convexity
x
x2
y2
x1
Preferences are strictly convex
when all mixtures z
are strictly
z
preferred to their
component
bundles x and y.
y
y1
38
Weak Convexity
Preferences are weakly
convex if at least one
mixture z is equally
preferred to a
component bundle.
x’
z’
x
z
y
y’
39
Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
40
More Non-Convex Preferences
x2
The mixture z
is less preferred
than x or y.
z
y2
x1
y1
41
Slopes of Indifference Curves
The slope of an indifference curve is its
marginal rate-of-substitution (MRS).
 The MRS measures the rate at which the
consumer is just willing to substitute one good
for the other.

42
Marginal Rate of Substitution
x2
MRS at x’ is the slope of the
indifference curve at x’
x’
x1
43
Marginal Rate of Substitution
x2
Dx2
x’
MRS at x’ is
lim {Dx2/Dx1}
Dx1
0
= dx2/dx1 at x’
Dx1
x1
44
Marginal Rate of Substitution
x2
dx2
x’
dx1
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.
x1
45
MRS & Ind. Curve Properties
Good 2
Two goods
a negatively sloped
indifference curve
MRS < 0.
Good 1
46
MRS & Ind. Curve Properties
Good 2
One good and one
bad
a positively
sloped indifference curve
MRS > 0.
Bad 1
47
MRS & Ind. Curve Properties
Good 2
MRS = - 5
MRS always increases (decreases in
absolute value) with x1 (becomes less
negative) if and only if preferences are
strictly convex.
MRS = - 0.5
Good 1
We call it a diminishing marginal rate of substitution.
48
MRS & Ind. Curve Properties
x2
MRS = - 0.5
MRS decreases
(becomes more negative)
as x1 increases with nonconvex
preferences.
MRS = - 5
x1
49
MRS & Ind. Curve Properties
x2
MRS is not always increasing as x1
increases with nonconvex preferences.
MRS = - 1
MRS
= - 0.5
MRS = - 2
x1
50