Question

please explain why

Suppose the Xavier values each apple at $1 and each banana at $2; Yolanda values each apple at $2 and each banana at $4; and Zoe value each apple at $2 and each banana at $1. Consider the following three allocations: Allocation 1: Xavier has 4 apples and 3 bananas, Yolanda has 2 apples and 2 bananas, and Zoe has 3 bananas and 4 apples. Allocation 2: Xavier has 3 apples and 4 bananas, Yolanda has 2 apples and 2 bananas, and Zoe has 4 bananas and 3 apples. Allocation 3: Xavier and Zoe have nothing, Yolanda has 9 apples and 9 bananas. Rank the three allocations according to the Pareto criterion. Allocation 1 Pareto dominates Allocation 2, but no other pair of allocations can be ranked No allocation Pareto dominates another. Allocation 3 Pareto dominates Allocation 1, but no other pair of allocations can be ranked. Allocation 1 Pareto dominates Allocation 2 and Allocation 3, and Allocation 3 Pareto dominates Allocation 2. Allocation 2 Pareto dominates Allocation 3, but no other pair of allocations can be ranked.

Answer 1

We make a table to explain the mechanism of comparison.

	Value for Apple	Value for Banana	Allocation 1	Value of allocation 1	Allocation 2	Value of Allocation 2	Allocation 3	Value of Allocation 3
x	1	2	(4,3)	10	(3,4)	11	(0,0)	0
y	2	4	(2,2)	12	(2,2)	12	(9,9)	54
z	2	1	(3,4)	10	(4,3)	11	(0,0)	0

Definition of Pareto-Improvement says that when the utility of any one agent can be improved without reducing the utility of others then we say that there is a Pareto-Improvement.

After this definition we can say that we can not compare Allocation 3 with any other Allocation. This is because utility of both x and z has reduced a lot in Allocation 3 in the process of increasing the utility of y. Hence Allocation 3 can not be compared.

Allocation 1 and Allocation 2 are comparable. Allocation 2 looks like a Pareto-Improvement over Allocation 1 because if we change allocation from Allocation 1 to Allocation 2 then we see that while y's utility remain constant there is improvement in utility of x and z.

Hence we can say that Allocation 2 is a Pareto-Improvement over Allocation 1.