ECON 7030 Microeconomic Analysis
Lecture 2: Preference
University of Queensland
Semester 1, 2023
Outline
Previous Lecture
• Consumption Set
• Consumption Bundles
• Budget Set
• Budget Line
This Lecture
• Binary Relation
• Preference
• Representation of preference
Part I
Preference
Preference
• Now that we have the consumption set, we can ask: how does
the consumer compares between pairs of consumption bundles
• In other words, if I give the consumer bundle A and bundle B,
will she
• prefers A to B, or
• prefers B to A, or
• likes them equally, or
• cannot compare?
• These comparisons are called Preference
Part II
Relation
Binary Relation
• A light switch (on/ off) is a good example of binary.
• Let R denote a binary relation
• xRy is called x in relation to y .
• x is a brother of y .
• x is richer than y .
• x is as good as y .
Binary Relations
• The technical term for two-way comparisons is Binary
Relations
• They are “binary” because you compare two at a time.
• They are “relations” because they can be compared.
Can you spot all binary relations on this slide?
1 Bluey: “Bingo’s watermelon is redder than mine!”
2 My child: “ I am healthier than my sister because I eat less
junk food.”
3 I did worse than you.
4 2 + 2 = 4 ≥ pi ≈ 3.1415927
Preference as a Binary Relations
• Preference is also a binary relation
• For example, I may say “Baby weakly prefers the orange bowl
to the blue bowl” if and only if she likes the orange bowl as
least as much as the blue bowl
(“weakly” because she may like them equally)
Note: a bowl is a bundle of pears and tomatoes.
• Since writing “is weakly preferred to” is cumbersome, we can
write in symbols:
Orange bowl R Blue bowl
Orange bowl%Blue bowl
which is the same as “the orange bowl is weakly preferred to
the blue bowl”
Two Derived Relations
• From the “weakly preferred to” relation, we can derived two
more binary relations:
Strictly preferred to
Bundle A is strictly preferred to Bundle B if and only if A
is weakly preferred to B but B is not weakly preferred to A
Or, in symbols:
A B if and only if A % B and B 6% A
Indifferent to
Bundle A is indifferent to Bundle B if and only if A is
weakly preferred to B and B is also weakly preferred to A
Or, in symbols:
A ∼ B if and only if A % B and B % A
Assumptions on Preference Relations
• So far, we have only required preferences to be binary relations
• But this being a course on economics (rather than
mathematics), we want to impose some more (economic)
assumptions on the preference relation
• These assumptions are sometimes known as axioms
Completeness
Name of the Axiom Completeness
Meaning Everything can be compared
Definition (in English)
For any two bundles A and B, I can COMPARE either
1 A is weakly preferred to B, or
2 B is weakly preferred to A, or
3 both
REMARK:
xRx or x is in relation to x itself is called Reflexive.
Mathematical Definition
For any two bundles A and B in the consumption set,
either A % B or B % A
An Example of Complete Preference
• Suppose instead Baby has a different preference in which she
weakly prefers Bowl A to Bowl B whenever Bowl A contains
as least as much food (either pear or tomato would do) as
Bowl B
• Mathematically, if Bowl A = (pA, tA) and Bowl B = (pB , tB),
then
A % B whenever pA + tA ≥ pB + tB .
• This preference relation is complete because either Bowl A
has as least as much food as B (in which case A % B), or B
has more food than A (in which case B % A)
• Mathematically,
• Either pA + tA ≥ pB + tB , in which case A % B;
• Or pA + tA ≤ pB + tB , in which case B % A
An Example of Complete Preference
A % B if and only if pA + tA ≥ pB + tB .
Bowl A = (3, 1)
Bowl B = (1, 3)
• pA + tA = 3 + 1 = 4
• pB + tB = 1 + 3 = 4
• 4 ≥ 4 ⇒ A % B
A Preference that is Not Complete
• Suppose Baby weakly prefers Bowl A to Bowl B only when
Bowl A contains more of both food than Bowl B
• This preference is not complete. Consider the following case:
Bowl A = (3, 1)
Bowl B = (1, 3)
• Neither is Bowl A weakly prefers to B (since A has fewer
tomato than B)
• Nor is Bowl B weakly prefers to A (since B has fewer pear
than A)
• So it is not true that either A % B or B % A. The preference
is not complete
Transitivity
Name of the Axiom Transitivity
Meaning The comparison is acyclic
Definition (in English)
For any three bundles A, B and C ,
if A is weakly preferred to B, and B is weakly preferred to C
then A is weakly preferred to C
Mathematical Definition
For any three bundles A, B and C in the consumption set,
A % B and B % C implies A % C
A Preference that is Transitive
• Again uppose Baby weakly prefers Bowl A to Bowl B
whenever Bowl A contains as least as much food (either pear
or tomato would do) as Bowl B, that is,
A % B whenever pA + tA ≥ pB + tB .
• This preference is transitive:
• Suppose A % B and B % C
• This means
pA + tA ≥ pB + tB (A % B)
≥ pC + tC (B % C )
• Hence A % C (since pA + tA ≥ pC + tC )
A Preference that is Transitive but Not Complete
• Again suppose Baby weakly prefers Bowl A to Bowl B only
when Bowl A contains more of both food than Bowl B
• This preference is also transitive:
• Suppose A % B and B % C
• This means
A has more of both food than B
B has more of both food than C
• So it must be that
A has more of both food than C
• Therefore A % C and the preference is transitive
Continuity
Name of the Axiom Continuity
Meaning No sudden reversal of
preferences
Idea Let A B C .
• Suppose I take a path from A
to C
• Along the path, I cannot
jump from strictly preferring
my position on the path to B
to strictly preferring B to my
position.
• There must be a point of
indifference in between
A
B
C
strictly prefers to B
strictly worse than B
indifferent to B
Monotonicity
Name of the Axiom
Monotonicity
Meaning More is better
Definition If bundle A has
strictly more of every good
than bundle B, then A B Pear
Tomato
Bundles in the shaded
area (excluding borders)
are strictly preferred to A
A
Strong Monotonicity
Name of the Axiom
Strong Monotonicity
Meaning More is better
Definition If bundle A has at
least as much of every good
than bundle B, and more of
at least one good, then
A B
Pear
Tomato
Bundles in the shaded
area (including borders)
are strictly preferred to A
A
Monotonicity VS Strong Monotonicity
Pear
Tomato
orange bowl
green bowl
blue bowl
pink bowl
• If preference satisfies Strong Monotonicity (and hence also
Monotonicity):
• Green bowl % Blue bowl
• Pink bowl % Orange bowl
• If preference satisfies Monotonicity but not Strong
Monotonicity:
• Green bowl % Blue bowl
Convexity: Averages Don’t Hurt
Name of the Axiom Convexity
Meaning Averages are not
worse than extremes
Definition If A % B, then an
average of A and B is also
weakly preferred to B Pear
Tomato
A
B
Bundles on the line are
weakly preferred to B
Strong Convexity: Averages Are Better
Name of the Axiom
Strong Convexity
Meaning Averages are better
than extremes
Definition If A % B, then an
average of A and B is
strictly preferred to B
Pear
Tomato
A
B
Bundles on the line are
strictly preferred to B
Convexity VS Strong Convexity
Pear
Tomato
A
B
• Suppose A % B
• If preference satisfies Strong Convexity (and hence also
Convexity):
• Bundles on the line are strictly preferred to B
• If preference satisfies Convexity but not Strong Convexity:
• Bundles on the line are weakly (but not necessarily strictly)
preferred to B
Defining Averages mathematically
Suppose A = (2, 4) and
B = (4, 2), then points in the
line are, for instance,
C =
1
2
A +
1
2
B
=
1
2
(2, 4) +
1
2
(4, 2)
= (1, 2) + (2, 1)
= (3, 3)
Generally, instead of 12 , the
weights on A and B might be
λ and 1− λ with 0 < λ < 1:
C = λ(2, 4) + (1− λ)(4, 2)
Pasta
Wine
A
B
Bundles on the line
strictly between A and B
are “strict averages.”
C
Excercise
Recall the preferences:
A % B whenever pA + wA ≥ pB + wB .
Are these preferences convex? Why?
Yes, for instance, A = (4, 1) is weakly preferred to B = (3, 2):
pA + wA = 4 + 1 = pB + wB = 3 + 2
For any λ ∈ (0, 1),
C = λ(4, 1) + (1− λ)(3, 2)
= (4λ, λ) + (3− 3λ, 2− 2λ)
= (3 + λ, 2− λ)
And pC + wC = 3 + λ+ 2− λ = 5.
Thus A % B and C % B.
Excercise
Recall the preferences:
A % B whenever pA + wA ≥ pB + wB .
Are these preferences strongly convex? Why?
No, in the previous slide we showed that C is not strictly preferred
to A (or B) even though
C =
1
2
A +
1
2
B
Part III
Representation of Preference
Making Life Easier
• Binary relations are cumbersome: we can only compare two at
a time
• Say we wish to find the tallest person in this course, with the
“taller than” comparison, we will need to make many
comparisons before being able to sort everyone
• It would be easier if I can simply ask for your height, and then
pick the one with the largest number
Representation
• In the example just now, height is a representation of the
“taller than” comparison in the following sense:
Height of Person A ≥ Height of Person B
if and only if
Person A is taller than Person B
• In other words, the “taller than” comparison is translated into
a comparison of numbers (height)
Utility Representation
• In the same manner we wish to have a representation of a
preference relation
• We call this representation Utility
• Utility is an assignment of numbers to each outcome, such
that
utility of x︷︸︸︷
u(x) ≥
utility of y︷︸︸︷
u(y)
if and only if
x % y
Can it be done?
Pear
Tomato
orange bowl
green bowl
blue bowl
pink bowl
• Even in a two-good world, the consumption set is
two-dimensional
• Utility, being numbers, live on a one-dimensional line
• It is not obvious that we must be able to find a utility
representation (which orders bundles in a two-dimensional
plane onto a one-dimensional line)
Existence of a Utility Representation
Theorem
If a preference relation is complete, transitive and continuous, then
it has a utility representation.
• As a matter of fact, if the consumption set is finite, only
completeness and transitivity is needed
• In such cases, rationality is often defined as preference being
complete and transitive
• Continuity is required only to deal with the dimensionality
problem mentioned in the last slide
A Note about Utility
• As you can see, Utility is a representation of preference, and
just that
• It doesn’t necessarily mean “happiness” or “satisfaction”
(These can be interpretations of utility in certain contexts)
Utility is Ordinal
• Recall that preference is about comparison only, there is no
content on magnitude
• As utility is a representation of preference, only comparison
should matter, magnitude should not
• That is, when A is preferred to B, we only required u(A) to
be bigger than u(B); we don’t care by how much
• In jargons, we say that “Utility is Ordinal” — which in plain
English means “only comparison matters”
A Mathematical Fact for our use
• Suppose f is a function that takes real numbers and outputs
real numbers
• f is strictly increasing if
x > y implies f (x) > f (y)
x = y implies f (x) = f (y)
x < y implies f (x) < f (y)
• Examples of strictly increasing functions: f (x) = 2x + 1,
f (x) = x3, f (x) = ln x
• Examples of functions that are not strictly increasing:
f (x) = 0, f (x) = x2 (if x can be negative)
Utility Representation is Not Unique
• As utility is ordinal, we can come up with different utility
representations for the same preferences
• In particular, if u is a utility function representing a preference
%, and f is a strictly increasing function, then we can define
the function v by
v(A) = f (u(A)) for any bundle A
and v will also represent the same preference %
• Why? Take two bundles A and B
A % B ⇐⇒ u(A) ≥ u(B) (u represents %)
⇐⇒ f (u(A)) ≥ f (u(B)) (f is strictly increasing)
⇐⇒ v(A) ≥ v(B) (Definition of v)
Illustration of Increasing Transformation
x
f (x)
u(x) = 0.6x
v (u(x)) = 1.4u(x) + 1
Summary
1 Preference is a binary relation over the consumption set
2 We can impose assumptions on preferences
Completeness Every pair of bundles can be compared
Reflexive Every bundle can be compared to itself
Transitivity The comparison is acyclic
Continuity No sudden reversal of preference
Monotonicity More is better
Convexity Averages do not hurt
3 When preference is complete, transitive and continuous, it has
a utility representation
4 If a preference has a utility representation, any increasing
transformation of such utility function also represents the
same preference.
5 Utility is ordinal — only comparison matters
Next Lecture
• Using utility for economic analysis