Rational Choice

Question 1

What is a Preference?

Answer 1

A Preference is a RELATION
- Binary Relation - between 2 entities
Small letters = Entities
Capital Letter = Relations
e.g. gRn - Germany is Bigger than Norway

Question 2

What is a Universe?

Answer 2

Specifies what sort of Entities may be Related

e.g. X = {Germany, UK, France, …} if comparing European countries

Question 3

What is a Preference Relation?

Answer 3

Relation to show Preferences between entities
e.g. ≤ and ≥ or ~
≥ is a Binary Relation on a Set of Alternatives, X (Universe)

Question 4

How do you show Preference Relations for specific individuals?

Answer 4

Using subscripts
e.g. c ≥(b) t - c is preferred to t by Betsy
t ≥(a) a - t is preferred to c by Alfred

Question 5

What other Relations can be derived from Binary Relation ≥?

Answer 5

1. Strict Preference Relation: > : x>y <=> x≥y but NOT y≥x

2. Indifference Relation: ~ : x~y <=> x≥y AND y≥x

Question 6

What is a Rational Preference Relation?

Answer 6

A Pref. Relation that is COMPLETE + TRANSITIVE

Question 7

What is Transitivity in terms of Relations?

Answer 7

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

Question 8

What is Completeness in terms of Relations?

Answer 8

R is Complete if:

• For ANY x and y in Universe, xRy or yRx or Both
• • Cannot fail to have Preferences between entities

Question 9

When can we Order Preferences?

Answer 9

When assuming Rational Preferences

Question 10

In Pref. Ordering, what does Completeness guarantee?

Answer 10

Completeness guarantees there is ONLY 1 Ordering

Question 11

In Pref. Ordering, what does Transitivity guarantee?

Answer 11

No Cycle in Strict Preferences

Question 12

What does Pref. Ordering allow us to do?

Answer 12

Assign Utility to entities

Question 13

What is the Standard Model of Choice Under Uncertainty?

Answer 13

• Consumer allocates Limited Income among diff. Consumption categories
  p(x) x + p(y) y ≤ m (BUDGET SET)

Question 14

What is a Consumption Set (menu)?

Answer 14

Space of ALL possible Bundles of G+S

Question 15

How can we make Rational Choices?

Answer 15

• Have Rational Pref. Ordering

- When faced w/ a Menu- choose most preferred item/one of most preferred items

Question 16

What is the Eq. Condition under Rational Choice Assumptions?

Answer 16

MRS xy = P(x) / P(y)

Question 17

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?

Answer 17

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

Question 18

What is Rationalisability?

Answer 18

Choice Rule is Rationalisable if there is a Preference ≥ such that C(A) = max(A, ≥)

Question 19

What is Revealed Preference Analysis?

Answer 19

1. Test Rational Choice - Consumer chooses most preferred Bundle
2. Discover Consumer’s Pref. Relation

Question 20

X = {a, y , z} and B = {x, y}
Given Choice Rule: C(X) = {x} and C(B) = {y}
Is this Choice Rationalisable?

Answer 20

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

Question 21

What is Direct Pref. Revelation under Rationalisability?

Answer 21

When chosen bundle x* is Revealed Directly as preferred to bundle y
- x >(D) y

Question 22

What is Indirect Pref. Revelation under Rationalisability?

Answer 22

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

Question 23

How do you reset Rationalisability w/ Choice Data?

Answer 23

WARP: Weak Axiom of Revealed Preference

Question 24

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?

Answer 24

If x is Chosen from A, then x must also be Chosen from B. But y is Chosen from B
- So, WARP NOT satisfied

Question 25

What is Equivalence Relation?

Answer 25

Relation R on a set X is:

1. Reflexive: if for ALL xϵX, xRx - at least as good as itself
2. Symmetric: if for ALL x,y ϵ X, xRy => yRx - e.g. same age as –> x~y

Question 26

How can Utility be used for orderings?

Answer 26

Assuming Completeness –> Utility func. can assign number w/ each element in Universe
=> x≥y <=> u(x) ≥ u(y)

Question 27

What is Representation Theorem?

Answer 27

If Universe is Finite + ≥ is Comp. + Transitive

- There exists a Utility Func. representing ≥