Deze samenvatting Behavioural Finance aan de KU Leuven bevat alle lessen van Prof Goncharenko en Prof Verdickt. Dit vak is nieuw sinds academiejaar 23-24.
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,PART 1
EXAM
NO QUESTIONS ON EXAM ABOUT MATHEMATICAL PRELIMINARIES: IT’S JUST FOR LATER ON SO WE
UNDERSTAND THE NEW CONCEPTS
Part 1: theory for problem solving: applications of theory
- Prep: slides, lectures, exercises in the book → book=!, do not rely on slides only
- 12 questions: 6 easy, 3 difficult, 3 very difficult (using concept, apply & be creative)
Part 2: will take questions from the book and put them in the exam
No theory, there will be exercises
LECTURE 1 CHOICE UNDER CERTAINTY
OUTLINE
- Mathematical preliminaries: Relation xRy
- Preference relation
- Rational preferences
- Choice under certainty
- Utility function
- Opportunity costs
- Sunk costs
- Menu dependence and the decoy theory
- Loss aversion and the endowment effect
- Anchoring and adjustment
MATHEMATICAL PRELIMINARIES
SETS
Formally, a set is a collection of non-identical elements
- for example consider set A = {a1, a2, a3}, since a2 is an element of A
we write a2 ∈ A
- Consider set A𝘫 = {a1, a2}; since every element of A𝘫 is also found in A, we say that A𝘫 is a
subset of A and write A𝘫 ⊂ A
1. Consider set B = {b1, b2, a3}; we define the intersection of sets A and B as
A ∩ B = {x : x ∈ A and x ∈ B}
- from the definition, it follows that A ∩ B = {a3} since a3 is the only common element of A and
B
2. We define the union of sets A and B as
A ∪ B = {x : x ∈ A or x ∈ B}
- from the definition, it follows that A ∪ B = {a1, a2, a3, b1, b2}
1
,SETS AND CARTESIAN PRODUCT
Let A and B be two non-empty sets, the Cartesian product of A and B is defined as
A × B = {(a, b) : a ∈ A and b ∈ B}.
Let A = {a1, a2} and B = {b1, b2}, then from the definition
A × B = {(a1, b1), (a2, b1), (a1, b2), (a2, b2)}
All possible combinations of A and B
(convince yourself that, in general, A × B /= B × A)
the Cartesian coordinate system is an example of the Cartesian product (multiplication of 2 sets)
RELATION XRY
- A (binary) relation on a nonempty set X is a subset R ⊆ X × X
- We write: xRy ⇐⇒ (x, y) ∈ R
Example:
If X = {x1, x2, x3} then X × X has 32 = 9 elements. Let’s define R such that R = {(x1, x1), (x2, x2), (x3, x3)}.
Clearly, R ⊆ X × X and thus defines a relation on X .
alternatively and equivalently, we can write x1Rx1, x2Rx2, and x3Rx3
Note, however, that given R in our example, there is no relation between any two different elements
on X . So, any idea what this relation actually is (you all know it but might not recognize at this
moment)?
Let R be a relation on X and denote by x , y , z generic elements of X . We say that R is
- reflexive if xRx for all x ∈ X
- symmetric if xRy implies yRx
- antisymmetric if xRy and yRx implies x = y
- transitive! if xRy and yRz implies xRz
- complete if for any x, y ∈ X either xRy or yRx (or both)
Example: Let X = {1, 2, 3} and R = {(1, 2), (1, 3), (2, 3)}. Which properties does this relation satisfy? Do
you recognize this relation?
PREFERENCE RELATION
Let X be a set of alternatives. For example, think of alternative ways of traveling to Brussels from
Leuven, in which case X = {bike, bus, car, train}.
Our weak preference relation on the set of alternative X is denoted by symbol ≥ (before we used a
generic R). For any two elements (alternatives) x, y ∈ X if we write
x≥y
2
, we mean that "x is at least as good as y ."
There are two more preference relations that we can derive from ≥:
The strict preference relation >.
- If x > y , we have x ≥ y , but not y ≥ x . We read x > y as "x is preferred to y."
The indifference relation ~.
- If x ~ y , we have x ≥ y and y ≥ x . We read x ~ y as "x is indifferent to y."
RATIONAL PREFERENCE RELATION
(under certainty)
The preference relation ≥ on the set of alternatives X is rational if it is:
(1) complete: for all x, y ∈ X either x ≥ y or y ≥ x (or both, you should be able to rank them)
(2) transitive: for all x, y, z ∈ X if x ≥ y and y ≥ z then x ≥ z
This may not seem like asking too much. However, haven’t you ever found yourself in a situation
when you simply cannot decide between two alternatives? If you have, then this is example of
preference incompleteness. Likewise, young children sometimes may exhibit intransitive preferences.
PRACTICE EXERCISE
To get yourself better familiar with preference relation, prove the following statements:
If the preference relation ≥ is rational, then
(1) > is both irreflexive ( x > x never holds) and transitive
(2) ~ is reflexive, transitive, and symmetric
(3) if x > y ≥ z , then x > z
UTILITY FUNCTION
Ideally, we would like to describe preference relations ≥ on a set of alternatives X by means of a
utility function. The reason is simple: mathematically, it is much easier (and efficient) to work with
functions than with binary relations. Thus, our utility function, u(x ), will assigns a numerical value to
each element x ∈ X , thereby ranking the elements of X in accordance with the individual’s
preferences.
A function u : X → R is a utility function representing preference relation
≥ if for all x, y ∈ X
Note: If such a function exists, then we can work with utility function instead of preferences. The
existence of such a function, in general, is rather hard to prove (and is far beyond the scope of this
course) and is required a couple of more technical assumptions on our preferences ≥. What you can
try to prove, however, is that if for any ≥ there exists u : X → R such that for all x, y ∈ X x ≥ y ⇐⇒ u(x )
≥ u(y ), then ≥ must be rational.
b ≥ 0 → u(b) ≥ u(0)
0 ≥ p → u(0 )≥ u(p)
U(b) ≥ p → u(b) ≥ u(p)
ADDITIONAL ASSUMPTIONS ON PREFERENCES
The consumer-choice theory under certainty is for practical purposes often supplemented by various
additional assumptions.
(1) non-satiation: given any element in x ∈ X close to it there is y ∈ X
such that u(y ) ≥ u(x ) (“more is always better”)
(2) convexity: for any x, y, z ∈ X and for any α ∈ [0,1] if x ≥ y and
z ≥ y then αx + (1 − α)z ≥ y (weighted average of x & z >= y)
(3) continuity: there are no "jumps" in preferences—no "lexicographic" preferences
3
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