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you just have to use the definition of pareto optimality here.
a pareto optimal allocation is one where for making a person better off, at least one other person is to be made worse off .
1)If alt x is chosen;
1, 4 and 7 are @ their most preferred positions.
to make ind 3 better off, say w is considered, but that will make one worse off .
also, 4 gets to his least preferred alt and hence is worse off.
hence, it may appear that x is pareto optimal.
but if alt y is considered, 1,3, 4,6 and 7 get same utility as before.
ind 5 gets better off..
therefore ind 5 is made better off without anyone made worse off. Thus, x is not pareto optimal.
2)If alt y is chosen.
1,2,4,5 & 7 all get their first choice alt..
but to make ind 3 or 6 better off, if any other alternative say w is considered it will make 1 and 5 ind worse off.
if x is considered here, then no one is getting better off in going from y to x, thus no point considering x.
if z is considered, ind 3 is getting better off in going from y to z, but that makes ind 1 worse off.
thus on deviating from y , every other alt satisfies criterion of pareto optimality..
hence, y is pareto optimal.
3) If alt z is chosen;
ind 3 and 6 have preferred choice. To make ind 1 better off , x can be considered but that will make ind 6 worse off.
if y is considered, ind 5 is getting better in going from z to y but that makes ind 6 worse off.
if w is considered, no one prefers w over z, therefore no point considering it as no one is getting better off.
thus, z is also pareto optimal.
4)If alt w is chosen:
w is no one's most preferred choice.
if w is choice,
then there exists an alternative 'z', that doesnt make anyone worse off.
all individuals 1-6 get better off.
ind 7 is indifferent between w and z, thus he is as well as before.
no one gets worse off!
thus w is not pareto optimal..
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