Preferences
Molly W. Dahl
Georgetown University
Econ 101 – Spring 2009
1
Rationality in Economics
Behavioral Assumption:
A decisionmaker always chooses the most
preferred alternative from the set of
available alternatives.
 To determine what a person chooses, we
must determine what a person prefers.

 That
is, to model choice we must model
preferences.
2
Preference Relations

Comparing two different consumption
bundles, x and y:
 strict

x is more preferred than is y.
 weak

preference
preference
x is as at least as preferred as is y.
 indifference

x is exactly as preferred as is y.
3
Preference Relations
Strict preference, weak preference, and
indifference are all preference relations.
 Particularly, they are ordinal relations

 they
state only the order in which bundles are
preferred.
4
Preference Relations
x
denotes strict preference
p
p

y means that bundle x is preferred
strictly to bundle y.
5
Preference Relations
x
denotes strict preference
p
p

y means that bundle x is preferred
strictly to bundle y.

f denotes weak preference
~f
x
y means x is preferred at least as much
~
as is y.
6
Preference Relations
x
denotes strict preference
p
p

y means that bundle x is preferred
strictly to bundle y.


f denotes weak preference
~f
x
y means x is preferred at least as much
~
as is y.
~ denotes indifference

x ~ y means x and y are equally preferred.
7
Preference Relations

f x imply x ~ y.
xf
y
and
y
~
~
8
Preference Relations
f x imply x ~ y.
xf
y
and
y
~
~

x
f y and (not y f x) imply x
~
~
p

y.
9
“Axioms” of Consumer Theory

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.
 Any
~
two bundles are comparable.
10
“Axioms” of Consumer Theory

Reflexivity: Any bundle x is always at
least as preferred as itself; i.e.
x
f x.
~
11
“Axioms” of Consumer Theory

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
f y and y f z
~
~
x
f z.
~
12
“Axioms” of Consumer Theory




Completeness
Reflexivity
Transitivity
If the Axioms of Consumer Theory hold, then
there will be an ordering of all alternatives.
 Preferences
are razor sharp.
13
Indifference Curves

Consider some bundle x’. An indifference
curve contains the set of all bundles
equally preferred to x’ (including x’ itself)
 That
is all bundles y such that y ~ x’.
14
Indifference Curves
x2
x’ ~ x” ~ x”’
x’
x”
x”’
x1
15
Indifference Curves
p
x
z
x
p
x2
y
z
y
x1
16
Indifference Curves
I1
x2
All bundles in I1 are
strictly preferred to
all in I2.
x
z
I2
y
I3
All bundles in I2 are
strictly preferred to
all in I3.
x1
17
Indifference Curves
x2
WP(x), the set of
x bundles weakly
preferred to x.
WP(x)
includes
I(x)
I(x).
x1
18
Indifference Curves
x2
SP(x), the set of
x bundles strictly
preferred to x,
does not
include
I(x)
I(x).
x1
19
Indifference Curves Cannot Intersect
x2
I1
I2 From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z.
x
y
z
x1
20
Indifference Curves Cannot Intersect
I1
I2 From I1, x ~ y. From I2, x ~ z.
Therefore y ~ z. But from I1
and I2 we see y z, a
contradiction.
x
y
p
x2
z
x1
21
Two “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.

22
Two “Goods”
Good 2
Two goods
a negatively sloped
indifference curve.
Good 1
23
A “Bad” and a “Good”

If less of a commodity is always preferred
then the commodity is a bad.
24
A “Bad” and a “Good”
Good 2
One good and one
bad
a
positively sloped
indifference curve.
Bad 1
25
Satiation

A bundle strictly preferred to any other is a
satiation point or a bliss point.
 Ice

cream and chocolate sauce
What do indifference curves look like for
preferences exhibiting satiation?
26
Indifference Curves Exhibiting Satiation
x2
Better
Satiation
(bliss)
point
x1
27
Indifference Curves Exhibiting Satiation
x2
Better
Satiation
(bliss)
point
x1
28
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 of the two commodities
in bundles determines their preference rankorder.
Blue or black pens
 Fiji Water or SmartWater

29
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
30
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.

Right and left shoes
31
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
32
Perfect Complements
x2
Since each of (5,5),
(5,9) and (9,5)
contains 5 pairs,
each is less
I2 preferred than the
bundle (9,9) which
I1 contains 9 pairs.
45o
9
5
5
9
x1
33
Well-Behaved Preferences

A preference relation is “well-behaved” if it
is both
 monotonic

and convex.
Monotonicity: More of any commodity is
always preferred (i.e. no satiation and
every commodity is a good).
 More
is better
34
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.
 Averages
are at least as good as extremes.
35
Well-Behaved Preferences -- Convexity
x
x2
x+y is strictly preferred
z=
2 to both x and y.
x2+y2
2
y
y2
x1
x1+y
1
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
Well-Behaved Prefs - 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).
 How can a MRS be calculated?

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’
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
44