23. Anita (A), Ben (B) and Carlos (C) are housemates who have moved to a new house and must decide how to allocate rooms X, Y and Z. An 'allocation' is where each housemate is assigned to exactly one room. For example, Anita → Room Z, Ben → Room X and Carlos → Room Y is allocation (Z, X, Y). Utilities for each room are given below: Room X 7 9 2 Room Y 4 3 7 Room Z 2 1 4 Utility for A Utility for B Utility for C (a) How many possible allocations are there in total? (b) Identify the two allocations which are not Pareto optimal and explain why they are not Pareto optimal. (Hint: is the allocation (Y,Z,X) Pareto optimal?) (c) Suppose we square Carlos' utility from each room (i.e. uc (Z) becomes 16). Would the set of Pareto optimal outcomes change? Why/why not? (d) Returning to the utilities from part (a), which of the Pareto optimal allocations maximise total surplus (utility) and would all housemates weakly prefer this allocation over any other? (e) Suppose the housemates decide to allocate the rooms by randomly assigning Room X to someone, Room Y to someone and Room Z to someone. Explain whether this always leads to an allocation which is Pareto optimal or maximises total surplus. (f) Suppose the housemates decide to allocate the rooms according to the 'random serial dictatorship' rule studied in lectures. Explain what this rule is and whether in this case it always leads to an allocation which is Pareto optimal or maximises total surplus.
23. Anita (A), Ben (B) and Carlos (C) are housemates who have moved to a new house and must decide how to allocate rooms X, Y and Z. An 'allocation' is where each housemate is assigned to exactly one room. For example, Anita → Room Z, Ben → Room X and Carlos → Room Y is allocation (Z, X, Y). Utilities for each room are given below: Room X 7 9 2 Room Y 4 3 7 Room Z 2 1 4 Utility for A Utility for B Utility for C (a) How many possible allocations are there in total? (b) Identify the two allocations which are not Pareto optimal and explain why they are not Pareto optimal. (Hint: is the allocation (Y,Z,X) Pareto optimal?) (c) Suppose we square Carlos' utility from each room (i.e. uc (Z) becomes 16). Would the set of Pareto optimal outcomes change? Why/why not? (d) Returning to the utilities from part (a), which of the Pareto optimal allocations maximise total surplus (utility) and would all housemates weakly prefer this allocation over any other? (e) Suppose the housemates decide to allocate the rooms by randomly assigning Room X to someone, Room Y to someone and Room Z to someone. Explain whether this always leads to an allocation which is Pareto optimal or maximises total surplus. (f) Suppose the housemates decide to allocate the rooms according to the 'random serial dictatorship' rule studied in lectures. Explain what this rule is and whether in this case it always leads to an allocation which is Pareto optimal or maximises total surplus.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:23. Anita (A), Ben (B) and Carlos (C) are housemates who have moved to a new house and
must decide how to allocate rooms X, Y and Z.
An 'allocation' is where each housemate is assigned to exactly one room. For example,
Anita → Room Z, Ben → Room X and Carlos → Room Y is allocation (Z, X, Y).
Utilities for each room are given below:
Room X
7
9
2
Room Y
4
3
7
Room Z
2
1
4
Utility for A
Utility for B
Utility for C
(a) How many possible allocations are there in total?
(b) Identify the two allocations which are not Pareto optimal and explain why they are
not Pareto optimal. (Hint: is the allocation (Y,Z) Pareto timal?)
(c) Suppose we square Carlos' utility from each room (i.e. uc (Z) becomes 16). Would
the set of Pareto optimal outcomes change? Why/why not?
(d) Returning to the utilities from part (a), which of the Pareto optimal allocations
maximise total surplus (utility) and would all housemates weakly prefer this
allocation over any other?
(e) Suppose the housemates decide to allocate the rooms by randomly assigning Room
X to someone, Room Y to someone and Room Z to someone. Explain whether this
always leads to an allocation which is Pareto optimal or maximises total surplus.
(f) Suppose the housemates decide to allocate the rooms according to the 'random
serial dictatorship' rule
in lect
Explain what this rule is and whether in this case it always leads to an allocation
which is Pareto optimal or maximises total surplus.
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