JNU - Pareto Optimal Alteratives

[Abhinav]

JNU 2011 - Ques 79

Suppose there are 3 alternatives x, y & z and four individuals 1,2,3 & 4. The individuals rankings (orderings) of the three alternatives R1, R2, R3, R4 are given by:

R1: (xy)z
R2: yzx
R3: z(xy)
R4: (xy)z

(Notation: Alternatives inside the parenthesis are indifferent to each other. If an alternative is written to the left of another alternative, then former is preferred to the latter.)

Then the set of Pareto-optimal alternatives is
(a) (x,y)
(b) (x,z)
(c) (y,z)
(d) None of the above

The answer in the solution is (c) (y,z)....I wanna know how to do it? In general how to do questions of Pareto optimality?

[duck]

Hi Abhinav.. :)
just use the definition of Pareto optimality.

Lets, check why "x" is not pareto optimal.
Consider a move from "x" to "y" ,  individuals 1,3,4 are indifferent. So, they are equally well of. And individual 2 is strictly better off. Therefore, "x" cannot be pareto optimal as we've found another alternative "y" which is making all indivduals equally well off and atleast one individual strictly better off.

Now, lets check for "y".
Consider a move from "y" to "x" , individuals 1,3,4 are indifferent but individual 2 is worse off.
Similarly, if we move from "y" to "z", Some individuals are better off and some are worse off.
So, there is no way in which we can make one person better off without making someone else worse off.Therefore, "y" is pareto optimal.

Now, lets check for "z".
Consider a move from "z" to "x", some individuals are better off and some are worse off.
Similarly, if we move to "y", some individuals are better off and some are worse off.
So, again there is no way in which we can make one person better off without makins someone else worse off. Therefore, "z" is also pareto optimal.

[Abhinav]

Thanks a lot...the discussion group is really helpful :)