Intended Learning Objectives:
a. Formal Rational Choice
b. Preferences and their Properties
c. Utility representation of preferences
d. Existence and (non)Uniqueness of Utility
Nicholson and Snyder: Chapter 3.
Varian: Chapters 3-4.
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Choice via Preferences
Consumer is Optimising: chooses the best given some constraints
Faced with two different consumption bundles x and y and no
constraints the consumer is able to indicate if one is better or not.
Preference: is a binary relation over objects of choice;
Choice set:
Bundle:
Binary:
for the consumer this is ℝ𝑛+ , where 𝑛 ∈ ℕ.
is denoted as 𝐱 = (𝑥1 , … , 𝑥𝑛 ), with 𝑥𝑖 being amounts
of good 𝑖 = 1, … , 𝑛. They are the objects of choice
means that two objects are compared
Symbols: For bundles in ℝ𝑛+ , we write:
Weak Preference:
𝐱 ≿ 𝐲 “𝐱 weakly preferred to 𝐲”
Strict Preference:
𝐱 ≻ 𝐲 “𝐱 strictly preferred to 𝐲”
Indifference:
𝐱 ~ 𝐲 “𝐱 indifferent to 𝐲”
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Utility
Analyst/Economist: model the optimising behaviour
This is done using mathematical tools and techniques.
Utility: is a function that represents the preference
A real - valued function U : R n  R represents the preference
of the consumer if for all bundles x, y  R n we have :
𝐱 ≿ 𝐲 if and only if 𝑈 𝐱 ≥ 𝑈 𝐲 .
We write 𝐱 ≿ 𝐲 ⇔ 𝑈 𝐱 ≥ 𝑈 𝐲
and call U a utility function of the consumer.
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Existence of a Utility
Gerard Debreu (1954): Preferences satisfying rationality and
continuity can be represented by a utility. This utility is ordinal.
Rational: Preferences are complete and transitive
A preference relation is complete if for all bundles 𝐱, 𝐲 ∈ ℝ𝑛+ we have:
𝐱 ≿ 𝐲 𝑜𝑟 𝐲 ≿ 𝐱 (or both ⇔ 𝐱 ~ 𝐲).
A preference relation is transitive if for all bundles 𝐱, 𝐲, 𝐳 ∈ ℝ𝑛+ we
have:
𝐱 ≿ 𝐲 and 𝐲 ≿ 𝐳 ⇒ 𝐱 ≿ 𝐳.
A preference relation is continuous if for all bundles 𝐱 ∈ ℝ𝑛+ the upper
contour set and the lower contour set are closed:
𝑈𝐶𝐱 ≔ 𝐲 ∈ ℝ𝑛+ : 𝐲 ≿ 𝐱 𝑎𝑛𝑑 𝐿𝐶𝐱 ≔ 𝐲 ∈ ℝ𝑛+ : 𝐱 ≿ 𝐲 are closed sets.
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The Intuition about Continuity
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Upper Contour set 𝑈𝐶𝐱
𝐱
𝑥2
indifferent
bundles
Lower Contour set 𝐿𝐶𝐱
𝑥1
1
A consequence of continuity is that indifference
sets “have no holes” however “fat” they may be.
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An Implication of Transitivity
2
we have : x ~ y
Suppose : y ~ z
Let’s add
continuity
Transitivi ty : x ~ z
𝐳
That is : U (x)  U (z )
𝑧2
𝐱
𝑥2
𝐲
𝑦2
UCx
indifferent
bundles
LCz
𝐿𝐶𝐳 ∩ 𝑈𝐶𝐱 ⊆ 𝐼𝐶𝐱 = 𝐼𝐶z .
𝐿𝐶𝐱 ∩ 𝑈𝐶𝐳 ⊆ 𝐼𝐶𝐱 = 𝐼𝐶z .
𝑥1 𝑧1 𝑦1
1
A consequence of transitivity is that different indifference curves cannot intersect.
Transitivity and continuity allow “fat” ICs: 𝑧1 > 𝑥1 , 𝑧2 > 𝑥2 but 𝒛 ~ 𝒙. IC x  IC z  IC y
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To exclude “fat” ICs: Monotonicity
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if 𝑧𝑖 > 𝑥𝑖 for all 𝑖 = 1, … , 𝑛 ⇒ 𝐳 ≻ 𝐱
strict monotonicity
𝐳
𝑧2
𝐱
𝑦2 = 𝑥2
𝐲
indifferent
bundles
𝑥1 𝑧1 𝑦1
1
Strong Monotonicity: if 𝑦𝑖 ≥ 𝑥𝑖 for all 𝑖 = 1, … , 𝑛, 𝐲 ≠ 𝐱 ⇒ 𝐲 ≻ 𝐱.
Weak Monotonicity: if 𝑦𝑖 ≥ 𝑥𝑖 for all 𝑖 = 1, … , 𝑛 ⇒ 𝐲 ≿ 𝐱.
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Now “easy” to find a Utility
2
45-degree
line
𝒖
𝐱
𝑥2
𝑢1𝐱 = 𝑢2𝐱
𝑦2
𝒖𝐱
𝐲
𝐲
indifferent
bundles
𝑥1 𝑢1𝐱
𝑦1 1
Slide on indifference curve 𝐼𝐶𝐱 until 45-degree line and find 𝒖𝐱 = 𝑈(𝐱)(1,1).
Slide on indifference curve 𝐼𝐶𝐲 until 45-degree line and find 𝒖𝐲 = 𝑈(𝐲)(1,1).
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How many Utility functions?
Ordinal Utility: means that any increasing transformation of a
utility also gives a utility function.
Proof : Suppose the consumers preference is rational and continuous .
Then there exist a utility U that represents preference .
𝐱 ≿ 𝐲 if and only if 𝑈 𝐱 ≥ 𝑈 𝐲 .
Let f : R  R be a strictly increasing function (i.e., a  b  f (a)  f (b)).
Then : U (x)  U (y )  f (U (x))  f (U (y )).
𝐱 ≿ 𝐲 if and only if 𝑓(𝑈 𝐱 ) ≥ 𝑓(𝑈 𝐲 ).
Therefore, a new utility, V  f (U ), represents preference .
Ordinal: only tells us that one bundle is better than another. It
does not enable us to make a statement by how much it is better.
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Essential Reading:
Nicholson and Snyder: Chapter 3.
Varian: Chapters 3-4.
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