2. Question The 'Democratic Family' consists of three members {mother m, father f, daughter d}, that have different preferences with respect to how much money X € [0,8] they think should be invested in the new family car. These preferences can be represented by the following utility functions: - Mother m: um(X) = 2X - X² Father f: uf(X) = 10X - X² - Daughter d: ud(X) = 4X - X² Calculate for each family member (f, m, d) the preferred amount of money to invest in the new car denoted by X; for i = {m, f,d}. Assume that (Xf, Xm, Xd) are the three alternatives on which the family must decide. Show that each family member has rational preferences over this domain. Preferences on the family level are determined by pairwise majority voting (all family members vote on two alternatives). Derive the family preference and check whether it is rational. Assume that the final family decision is made by conducting sequential pair- wise majority voting, where the loosing alternative is eliminated. Does the outcome depend on the voting sequence? If yes, provide two sequences that yield different results. If not, provide an argument why sequential pairwise majority voting does not depend on the voting sequence. Propose an alternative utility function for the daughter that results in a non- transitive family preference. Derive the corresponding family preference. As- sume that the mother decides about the voting sequence in c). What voting sequence would she implement given she is aware of the alternative utility function of her daughter. Justify your response and discuss its implications.
2. Question The 'Democratic Family' consists of three members {mother m, father f, daughter d}, that have different preferences with respect to how much money X € [0,8] they think should be invested in the new family car. These preferences can be represented by the following utility functions: - Mother m: um(X) = 2X - X² Father f: uf(X) = 10X - X² - Daughter d: ud(X) = 4X - X² Calculate for each family member (f, m, d) the preferred amount of money to invest in the new car denoted by X; for i = {m, f,d}. Assume that (Xf, Xm, Xd) are the three alternatives on which the family must decide. Show that each family member has rational preferences over this domain. Preferences on the family level are determined by pairwise majority voting (all family members vote on two alternatives). Derive the family preference and check whether it is rational. Assume that the final family decision is made by conducting sequential pair- wise majority voting, where the loosing alternative is eliminated. Does the outcome depend on the voting sequence? If yes, provide two sequences that yield different results. If not, provide an argument why sequential pairwise majority voting does not depend on the voting sequence. Propose an alternative utility function for the daughter that results in a non- transitive family preference. Derive the corresponding family preference. As- sume that the mother decides about the voting sequence in c). What voting sequence would she implement given she is aware of the alternative utility function of her daughter. Justify your response and discuss its implications.
Chapter3: Preferences And Utility
Section: Chapter Questions
Problem 3.9P
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![2. Question
The 'Democratic Family' consists of three members {mother m, father f, daughter
d}, that have different preferences with respect to how much money X € [0,8]
they think should be invested in the new family car. These preferences can be
represented by the following utility functions:
-
Mother m: um(X) = 2X - X²
Father f: uf(X) = 10X - X²
- Daughter d: ud(X) = 4X - X²
Calculate for each family member (f, m, d) the preferred amount of money
to invest in the new car denoted by X; for i = {m, f,d}. Assume that
(Xf, Xm, Xd) are the three alternatives on which the family must decide. Show
that each family member has rational preferences over this domain.
Preferences on the family level are determined by pairwise majority voting (all
family members vote on two alternatives). Derive the family preference and
check whether it is rational.
Assume that the final family decision is made by conducting sequential pair-
wise majority voting, where the loosing alternative is eliminated. Does the
outcome depend on the voting sequence? If yes, provide two sequences that
yield different results. If not, provide an argument why sequential pairwise
majority voting does not depend on the voting sequence.
Propose an alternative utility function for the daughter that results in a non-
transitive family preference. Derive the corresponding family preference. As-
sume that the mother decides about the voting sequence in c). What voting
sequence would she implement given she is aware of the alternative utility
function of her daughter. Justify your response and discuss its implications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Facdfefc0-f3e3-4692-8e86-7ea128e6487e%2Fe2125167-474f-475e-9892-1aa3a4462cc5%2Fh5lgiob_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Question
The 'Democratic Family' consists of three members {mother m, father f, daughter
d}, that have different preferences with respect to how much money X € [0,8]
they think should be invested in the new family car. These preferences can be
represented by the following utility functions:
-
Mother m: um(X) = 2X - X²
Father f: uf(X) = 10X - X²
- Daughter d: ud(X) = 4X - X²
Calculate for each family member (f, m, d) the preferred amount of money
to invest in the new car denoted by X; for i = {m, f,d}. Assume that
(Xf, Xm, Xd) are the three alternatives on which the family must decide. Show
that each family member has rational preferences over this domain.
Preferences on the family level are determined by pairwise majority voting (all
family members vote on two alternatives). Derive the family preference and
check whether it is rational.
Assume that the final family decision is made by conducting sequential pair-
wise majority voting, where the loosing alternative is eliminated. Does the
outcome depend on the voting sequence? If yes, provide two sequences that
yield different results. If not, provide an argument why sequential pairwise
majority voting does not depend on the voting sequence.
Propose an alternative utility function for the daughter that results in a non-
transitive family preference. Derive the corresponding family preference. As-
sume that the mother decides about the voting sequence in c). What voting
sequence would she implement given she is aware of the alternative utility
function of her daughter. Justify your response and discuss its implications.
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