Rational Choice Flashcards
What is a Preference?
A Preference is a RELATION - Binary Relation - between 2 entities Small letters = Entities Capital Letter = Relations e.g. gRn - Germany is Bigger than Norway
What is a Universe?
Specifies what sort of Entities may be Related
e.g. X = {Germany, UK, France, …} if comparing European countries
What is a Preference Relation?
Relation to show Preferences between entities
e.g. ≤ and ≥ or ~
≥ is a Binary Relation on a Set of Alternatives, X (Universe)
How do you show Preference Relations for specific individuals?
Using subscripts
e.g. c ≥(b) t - c is preferred to t by Betsy
t ≥(a) a - t is preferred to c by Alfred
What other Relations can be derived from Binary Relation ≥?
- Strict Preference Relation: > : x>y <=> x≥y but NOT y≥x
2. Indifference Relation: ~ : x~y <=> x≥y AND y≥x
What is a Rational Preference Relation?
A Pref. Relation that is COMPLETE + TRANSITIVE
What is Transitivity in terms of Relations?
R is Transitive if:
- For ALL x, y and z in Universe, if xRy and yRz then xRZ
- Need at least 3 alternatives in Universe
What is Completeness in terms of Relations?
R is Complete if:
- For ANY x and y in Universe, xRy or yRx or Both
- Cannot fail to have Preferences between entities
When can we Order Preferences?
When assuming Rational Preferences
In Pref. Ordering, what does Completeness guarantee?
Completeness guarantees there is ONLY 1 Ordering
In Pref. Ordering, what does Transitivity guarantee?
No Cycle in Strict Preferences
What does Pref. Ordering allow us to do?
Assign Utility to entities
What is the Standard Model of Choice Under Uncertainty?
- Consumer allocates Limited Income among diff. Consumption categories
p(x) x + p(y) y ≤ m (BUDGET SET)
What is a Consumption Set (menu)?
Space of ALL possible Bundles of G+S
How can we make Rational Choices?
- Have Rational Pref. Ordering
- When faced w/ a Menu- choose most preferred item/one of most preferred items
What is the Eq. Condition under Rational Choice Assumptions?
MRS xy = P(x) / P(y)
Assume agent given Set of Choices: A c X
- Agent will choose elements of A preferred the most
What is the Choice Rule w/ Relation ≥over X?
C(A, ≥) = {xεA: x≥y for all yεA}
- If no ambiguity over which Agent we are talking about - can just write C(A) - Choice from menu A c X
What is Rationalisability?
Choice Rule is Rationalisable if there is a Preference ≥ such that C(A) = max(A, ≥)
What is Revealed Preference Analysis?
- Test Rational Choice - Consumer chooses most preferred Bundle
- Discover Consumer’s Pref. Relation
X = {a, y , z} and B = {x, y}
Given Choice Rule: C(X) = {x} and C(B) = {y}
Is this Choice Rationalisable?
C(X) => x≥y nad x≥z –> x is Unique choice of X –> x>y and x>z
C(B) => y > x - This is NOT Transitive
– Can’t be Rationalised
What is Direct Pref. Revelation under Rationalisability?
When chosen bundle x* is Revealed Directly as preferred to bundle y
- x >(D) y
What is Indirect Pref. Revelation under Rationalisability?
Suppose x is Revealed Directly Preferred to y
- And y is Revealed Directly Pref. to z
By TRANSITIVITY - x is Revealed Indirectly Pref. to z
How do you reset Rationalisability w/ Choice Data?
WARP: Weak Axiom of Revealed Preference
If x,y ϵ AnB (x and y both affordable in Budget sets A and B)
Then if x ϵ C(A) and y ϵ C(B), does this satisfy WARP?
If x is Chosen from A, then x must also be Chosen from B. But y is Chosen from B
- So, WARP NOT satisfied
What is Equivalence Relation?
Relation R on a set X is:
- Reflexive: if for ALL xϵX, xRx - at least as good as itself
- Symmetric: if for ALL x,y ϵ X, xRy => yRx - e.g. same age as –> x~y
How can Utility be used for orderings?
Assuming Completeness –> Utility func. can assign number w/ each element in Universe
=> x≥y <=> u(x) ≥ u(y)
What is Representation Theorem?
If Universe is Finite + ≥ is Comp. + Transitive
- There exists a Utility Func. representing ≥
