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ECON 7110: Consumer and Firm Behaviour

Lecture 3: Preference and Utility

Allan Hernandez Chanto

University of Queensland

Semester 1, 2021

What is Economics?

Economics is the study of

I How decision makers CHOOSE

I Among limited POSSIBILITIES

I To satisfy WANTS (objectives, preferences,. . .

WANTS: Rational vs. Behavioural

I Rational { I DO what I WANT.

I Behavioral { I WANT what I DO.

I Either way, in economics we will use \Preferences” to describe

WANTS.

I We separate the WANTS (described by preferences) from

what is POSSIBLE (a_ordable, feasible).

Objects of Choice: Consumption Bundles

I A Consumption Bundle is a bundle of goods.

I In general, we have many goods: food, cars, houses,

computers, hair cuts, swimming sessions, psychiatric sessions,

dentist appointments, etc.

I A bundle of goods is described by a vector (5; 4; 4; 2; 1; 7).

I This bundle has 5 units of the _rst good.

I 4 units of the second good, etc.

I Let’s focus on a 2-good world.

I Why 2 goods?

I It takes 2 to express a trade-o_!

Allan’s 2-Good World

I Allan likes to consume two goods: Pasta and Wine.

I A consumption bundle is a pair (p;w), in which p denotes the

quantity of plates of pasta, and w the number of glasses of

wine in the bundle.

I Notice that each bundle can be completely described by 2

numbers: the amount of pasta and the amount of wine.

I However, order matters!

Table: Some Examples of Consumption Bundles

Pasta (units) Wine (units)

Bundle 1 1 2

Bundle 2 3 1

Bundle 3 2 0

Bundle 4 1 _

Representing Consumption Bundles – Cartesian Plane

Pasta

Wine

(1; 2)

(3; 1)

(2; 0)

(1; _)

Preferences over Consumption Bundles

I We can take the consumption set to be ALL possible bundles

of pasta and wine. That is, ALL pairs of non-negative real

numbers.

I Or, we could take it to be ALL pairs of non-negative integers.

Then the consumption set has \gaps in some sense.”

I It is sometimes convenient to use all pairs of non-negative

reals. NO GAPS! The whole non-negative cartesian plane.

Preferences over Consumption Bundles

I We are interested in how a consumer \values” di_erent

consumption bundles (\pairs of pairs” in a 2-good world, or

\pairs of vectors” in a many good world).

I If a consumer is confronted with bundle A and bundle B, she

may

I Prefer one over the other.

I Like them just as much, or

I Be unable to decide.

I These comparisons are called Preferences

Buridan’s Donkey Paradox

I Have you heard about Buridan’s Donkey Paradox? Let

Sheldon and Amy teach us the concept of the Buridan’s

Donkey|The Big Bang Theory

I Can you \_x” the Burdian’s Donkey Paradox using a

preference approach?

Cartesian Plane or Grid

Good 1

Good 2

I Using the Cartesian Plane means we can subdivide goods as

_ne as we want.

I Or we can use a Grid.

Binary Relations

I The technical term for two-way comparisons is Binary

Relations.

I E.g., \bigger than,” \cuter than,” \is the mother of.”

I They are called \binary” because they compare two objects

(bundles) at a time.

I They are called \relations” because they state whether or not

some relationship holds between the two objects.

I For example, Mary \is the mother of” Jane.

I Mary and Jane are the objects.

I “is the mother of” is the relationship.

I Bundle (4; 2) \is at least as good as” bundle (2; 4).

I (4; 2) and (2; 4) are the two objects.

I \is at least as good as” is the relationship.

Preference as a Binary Relation

I First, take the Consumption Set over which the preferences

are de_ned to be the Cartesian Plane (non-negative part).

I A \is at least as good as” B if A is better than B or A is just

as good as B.

I This is all according to the consumer (Consumer Sovereignty).

I This allows \indi_erence” – you like each just as much as the

other.

I Indi_erence is not to be confused with . . . \I don’t know” or \I

can’t compare them.”

Preference as a Binary Relation

I Since \is at least as good as” is already a long statement, we

have invented a new symbol %:

x % y

is short hand for \x is at least as good as y”

I This is similar but NOT EQUIVALENT to _.

Two Derived Relations

I 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 A is at least as

good as B, but B is not at least as good as A.

Or, in symbols:

A _ B if and only if A % B and B 6% A

Indi_erent to

Bundle A is indi_erent to Bundle B if A is at least as good

as B, and B is also at least as good as A.

Or, in symbols:

A _ B if and only if A % B and B % A

Assumptions on Preferences %

Name of Assumption Completeness

De_nition (in English)

For any two bundles A and B, either A is at least as good as B,

or B is at least as good as A.

Mathematical De_nition

For any two bundles A and B,

either A % B or B % A

Meaning Every pair of bundles can be compared.

I We use \or” to allow BOTH A % B and B % A |

Indi_erence.

I A and B as variables can even be the same bundle, for

example, A = (2,2) and B = (2,2).

An Example of Complete Preference

I Suppose Allan weakly prefers A to B whenever A contains as

least as much TOTAL food (plates of pasta plus glasses of

wine) as B.

I Mathematically, let A = (pA;wA) and B = (pB;wB), then

A % B whenever pA + wA _ pB + wB.

I This preference relation is complete.

I Mathematically,

I Either pA + wA _ pB + wB, in which case A % B;

I Or pA + wA < pB + wB, in which case B % A.

A Preference that is Not Complete

I Suppose A % B if and only if pA _ pB and wA _ wB (A has

at least as much pasta and wine as B).

I This preference is not complete. Consider the following

bundles:

A = (3; 1)

B = (1; 3)

I Neither A % B (since A has fewer glasses of wine).

I Nor B % A (since B has fewer plates of pasta).

Transitivity

Name of Assumption Transitive

De_nition (in English)

For any three bundles A, B and C, if A is at least as good as B,

and B is at least as good as C, then A is at least as good as C.

Mathematical De_nition

For any three bundles A, B and C,

A % B and B % C imply A % C

Meaning The relation can be \transferred.”

A Preference that Is Transitive

I A % B i_ pA + wA _ pB + wB.

I This preference is transitive:

I Suppose A % B and B % C

I This means

pA + wA _ pB + wB (A % B)

pB + wB _ pC + wC (B % C)

I Hence A % C (since pA + wA _ pC + wC )

I Notice, the “greater than or equal to” relation _ is transitive

on the reals.

Exercise 1

I Suppose Allan weakly prefers A = (pA;wA) to B = (pB;wB) if

the product of the consumption in A is greater than or equal

than the consumption in B

A % B whenever pA _ wA _ pB _ wB.

I Is this preference complete?

I Is this preference transitive?

Implications of Complete and Transitive Preferences

I For a _nite set of bundles, complete and transitive preferences

ensure that there is a optimal (\best”) choice according to

those preferences.

I A is best means A is at least as good as every other bundle.

I Note there may be more than one best choice, in which case

the individual is indi_erent between all the \best” ones.

I If % is transitive, then so are _ and _.

Comparison of % over Bundles to _ over Numbers

Both preferences % over bundles and _ over numbers are:

I Complete and transitive | Check for yourself.

I However, _ has another property that preferences might not

have . . .

I It is called anti-symmetry.

De_nition

If A % B and B % A, then A and B must be the same item

(bundle, number, etc.).

I Why do we allow for violations of anti-symmetry with

preferences?

I To allow indi_erence between di_erent bundles of goods x and

y.

More is Better (Weak Version)

Name of the Assumption

Monotonicity.

De_nition

If bundle A has more than

bundle B of every good, then

A _ B.

Meaning More (of ALL goods

together) is better.

Pasta

Wine

Bundles in the shaded

area (excluding borders)

are strictly preferred to

A.

A

More Is Better (Strong Version)

Name of the Assumption

Strong Monotonicity.

De_nition If bundle A has at

least as much as bundle B

of every good, and more of

at least one good, then

A _ B.

Meaning More (of ANY group

of goods) is better.

Pasta

Wine

Bundles in the shaded

area (including borders)

are strictly preferred to

A.

A

Di_erence between Monotonicity and Strong Monotonicity

Pasta

Wine

O

G

B

P

I If preferences satisfy Strong Monotonicity (and hence also

Monotonicity):

I G _ B

I P _ O

I If preferences satisfy Monotonicity but not Strong

Monotonicity:

I G _ B

Averages Don’t Hurt (Weak Version)

Name of the Assumption

Convexity.

De_nition If A _ B, then any

average of A and B is at

least as good as A.

Meaning Averages are not

worse than extremes.

Pasta

Wine

A

B

Bundles on the line be-

tween A and B are aver-

ages of A and B.

Averages Are Better (Strong Version)

Name of the Axiom

Strong Convexity.

De_nition If A _ B, then any

\strict” average of A and B

is better than A.

Meaning Averages are better

than extremes.

Pasta

Wine

A

B

Bundles on the line

strictly between A and B

are \strict averages.”

C

Exercise 1: Revisited

I Suppose Allan weakly prefers A = (pA;wA) to B = (pB;wB) if

the product of the consumption in A is greater than or equal

than the consumption in B

A % B whenever pA _ wA _ pB _ wB.

I Is this preference monotone?

I Is this preference strongly monotone?

Nearby Bundles Are Similar

Name of the Assumption

Continuity.

De_nition If A _ B, then

A0 _ B0 whenever A0 is

\su_ciently close” to A,

and B0 is \su_ciently

close” to B.

Meaning Small changes in

bundle sizes do not change

preferences.

Pasta

Wine

A

B

Utility Representation

I Binary relations are cumbersome.

I It would be easier to have a numerical representation of

preferences.

I We call such a representation Utility.

I Utility is an assignment of numbers to each bundle, such that:

u(x) _ u(y) if and only if x % y.

Existence of a Utility Representation

Theorem

If a preference relation is complete, transitive and continuous, then

it has a utility representation.

I As a matter of fact, if the consumption set is _nite (or

countably in_nite), only completeness and transitivity is

required to have a representation.

I Continuity is required to deal with the continuum (Cartesian

Plane).

I Other assumptions {convexity and monotonicity { are natural

for economics problems and have some other bene_ts.

Utility Is Ordinal

I Recall that preference is about comparison only, there is no

content on magnitude.

I So, . . . when A is at least as good as B, we only required u(A)

to be at least as large as u(B); we don’t care by how much

I We say that \Utility is Ordinal” | which in plain English

means \only order matters” (better, indi_erent, or worse), not

magnitude \cardinality.”

Real-life Application: College Admission Systems

I In many countries around the world students are assigned to

majors via a \centralised system.”

I Students are ranked according to their score in a standardized

exam.

I They are asked to report preferences over academic programs.

These preferences are ordinal.

I An algorithm assigns students to academic programs using the

reported preferences and scores.

I How can you compare the \intensity in preferences” of two

di_erent students?

Other Real-life Applications

I Votes in committees.

I Political elections.

I Preferences over courses.

I Preferences over rooms in students’ residences.

I Any other example?

A Mathematical Fact for Our Use

I Suppose f is a function that assigns a real number to each

real number.

I 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)

I Examples of strictly increasing functions: f (x) = 2x + 1,

f (x) = x2, f (x) = ln x, where the domain of x is positive real

numbers.

I Examples of functions that are not strictly increasing:

f (x) = 0, f (x) = 􀀀x2, where again the domain of x is the

positive real numbers.

Utility Representation Is Not Unique

I As utility is ordinal, we can come up with di_erent utility

representations for the same preferences.

I In particular, if u is a utility function representing a preference

%, and f is a strictly increasing function, then we can de_ne

the function v by

v(A) = f (u(A)) for any bundle A

and v will also represent the same preferences %.

Multiple Representations

I We stated Allan’s preferences over Pasta and Wine as A % B

i_ pA + wA _ pB + wB.

I A natural representation is: u(A) = pA + wA.

I Another one is: v(A) = 0:5[(pA)2 + (wA)2] + pAwA.

I Here f (x) = 0:5×2 which is strictly increasing on non-negative

reals.

I v(A) = f (u(A)) = 0:5[pA + wA]2.

I Con_rm it!

Summary

1. Preference is a binary relation over the consumption set.
2. We can impose assumptions on preferences.

Completeness Everything can be compared.

Transitivity The preference is transferable.

Continuity Close bundles are similar.

Monotonicity More is better.

Convexity Averages are better.

3. When preference is complete, transitive and continuous, it has

a utility representation.

4. Utility is ordinal | only order matters {better, worse, or

indi_erent . . .

Next Time

I More on working with utility in the consumption set.

I Also: Constraints | not everything in the consumption set is

feasible!

Tutorial 3: Consumer’s Choice

ECON 7110: Consumer and Firm Behaviour

The University of Queensland

Semester 1, 2021

Question 1

Three friends, Alice, Bob and Carol, have the following individual preferences over the food they

usually get: Chinese, Indian and Italian.

Alice Bob Carol

Most preferred Chinese Italian Indian

Indian Chinese Italian

Least preferred Italian Indian Chinese

Jointly, they agree to determine their group preference as follows: Cuisine A is preferred to B

if A gets the majority of votes against B in a two-alternative vote.

• What is the group’s preference over Chinese and Indian food?
• What is the group’s preference over Indian and Italian food?
• What is the group’s preference over Chinese and Italian food?
• Is the group’s preference over the three cuisines transitive? Why or why not?

Question 2

In a discussion of tuition rates, a university official argues that the demand for admission is

completely price inelastic. As evidence, she notes that while the university has doubled its

tuition (in real terms) over the past 15 years, neither the number nor quality of students applying

has decreased. Would you accept this argument? Explain briefly. (Hint 1: The official makes

an assertion about the demand for admission, but does she actually observe a demand curve?

What else could be going on? Hint 2: Think about the existence of potential substitutes.)

Question 3

Suppose there are only two goods: Beer and Milk. Tom’s preference over bundles of beer and

milk is as follows: For any two bundles A = (bA,mA) and B = (bB,mB) (where b and m denotes

the amount of beer and milk, respectively), A % B (i.e., “A is at least as good as B”) if and

only if:

Either bA > bB;

Or bA = bB and mA _ mB.

In other words, Tom cares, first and foremost, about the amount of beer, but if the two bundles

contain the same amount of beer, then he prefers having more milk to less.

• Is Tom’s preference complete? If yes, show why; if no, give an example of two bundles

between which Tom cannot compare.

• Is Tom’s preference monotone? Strongly monotone?
• Does Tom’s preference comply with the property of diminishing marginal rate of substitution?

2

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