(5) (Allais' Paradox) This example provides a natural preference order (natural meaning that a large percentage of people will agree with the proposed preference) and it provides a contradiction with the existence of a von Neumann-Morgestern representation of (and under the assumption of the independence property). Define the following lotteries Q1 = 0.33 $2,500+ 0.66 8$2,400+ 0.01 oso, P₁ = $2,400 P₂ = 0.34 882,400 +0.66 880, Q2 = 0.33 $2,500 +0.67 8so. Let us use X to denote, in a generic way, the payoff associated to the corresponding lottery (as explained in the notes); answer the following: • Check that EQ, (X) = $2,409 > EP, (X) = $2,400. • Check that Eq₂(X) = $825 >Ep₂ (X) = $816. . Most people (as per empirical experiments) have the following preferences P₁ > Q₁ (notice that this reverses the inequality between the corresponding expected values) and Q2P2 (notice that this is consistent with the inequality between the corresponding expected values). Use the property of independence together with P₁ Q₁ and Q2 > P2 to show that the following holds for all a € (0, 1): aP₁ + (1 -a)Q2aQ₁ + (1 -a)Q2aQ₁ + (1 -a) P₂. • Taking a = 1/2 in (**) derive a contradiction with the assumption that there exists an affine representation function U of>. Hint: to obtain the said contradiction look/derive an equality being satisfied by P₁, P2, Q1, Q2.
(5) (Allais' Paradox) This example provides a natural preference order (natural meaning that a large percentage of people will agree with the proposed preference) and it provides a contradiction with the existence of a von Neumann-Morgestern representation of (and under the assumption of the independence property). Define the following lotteries Q1 = 0.33 $2,500+ 0.66 8$2,400+ 0.01 oso, P₁ = $2,400 P₂ = 0.34 882,400 +0.66 880, Q2 = 0.33 $2,500 +0.67 8so. Let us use X to denote, in a generic way, the payoff associated to the corresponding lottery (as explained in the notes); answer the following: • Check that EQ, (X) = $2,409 > EP, (X) = $2,400. • Check that Eq₂(X) = $825 >Ep₂ (X) = $816. . Most people (as per empirical experiments) have the following preferences P₁ > Q₁ (notice that this reverses the inequality between the corresponding expected values) and Q2P2 (notice that this is consistent with the inequality between the corresponding expected values). Use the property of independence together with P₁ Q₁ and Q2 > P2 to show that the following holds for all a € (0, 1): aP₁ + (1 -a)Q2aQ₁ + (1 -a)Q2aQ₁ + (1 -a) P₂. • Taking a = 1/2 in (**) derive a contradiction with the assumption that there exists an affine representation function U of>. Hint: to obtain the said contradiction look/derive an equality being satisfied by P₁, P2, Q1, Q2.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Transcribed Image Text:(5) (Allais' Paradox) This example provides a natural preference order (natural meaning
that a large percentage of people will agree with the proposed preference) and it provides
a contradiction with the existence of a von Neumann-Morgestern representation of (and
under the assumption of the independence property). Define the following lotteries
Q1 = 0.33 $2,500+ 0.66 8$2,400+ 0.01 oso, P₁ = $2,400
P₂ = 0.34 882,400 +0.66 880, Q2 = 0.33 $2,500 +0.67 8so.
Let us use X to denote, in a generic way, the payoff associated to the corresponding lottery
(as explained in the notes); answer the following:
• Check that EQ, (X) = $2,409 > EP, (X) = $2,400.
• Check that Eq₂(X) = $825 >Ep₂ (X) = $816.
. Most people (as per empirical experiments) have the following preferences P₁ > Q₁
(notice that this reverses the inequality between the corresponding expected values) and
Q2P2 (notice that this is consistent with the inequality between the corresponding
expected values). Use the property of independence together with P₁ Q₁ and Q2 > P2
to show that the following holds for all a € (0, 1):
aP₁ + (1 -a)Q2aQ₁ + (1 -a)Q2aQ₁ + (1 -a) P₂.
• Taking a = 1/2 in (**) derive a contradiction with the assumption that there exists an
affine representation function U of>. Hint: to obtain the said contradiction look/derive
an equality being satisfied by P₁, P2, Q1, Q2.
AI-Generated Solution
AI-generated content may present inaccurate or offensive content that does not represent bartleby’s views.
Unlock instant AI solutions
Tap the button
to generate a solution
Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education