dse entrance 2012 question doubt

[kam]

this refers to question no 27, we have utility functions U^a=x+y and U^b=x^2 + y^2 and endowments are A(2,1) and B(1,2), two allocations are given, allocation 1-A(1.5,0) and B(1.5,3) and allocation 2-A(1.5,1.5) and B(1.5,1.5). The answer key says that allocation 1 is pareto optimal, but how can this be since A's utility is decreasing

[Sumit]

My take Acc.to definition of pareto efficiency : It's not only mean making one people better of without hurting anybody else..(this definition is used only when already efficiency is there but in this question it is not)....Its also mean to make some group of people better off...It's does not concern about distribution of utility (which you stuck off)...it look only for total utility Maximization...which is done only in first Allocation thats I think its  only Allocation-1 is pareto efficiency..

[Sumit]

I also hv little doubt in it....that we can get total utility(i.e A+B) maximized by allocating B(3,3)=18 which is greater then Allocation-1(i.e 12.75) in that case it should be...none of the allocation is pareto optimum...I think question take best of these allocation as pareto optimum....I think Amit sir will help us in this...

[Amit Goyal]

We have
U^a=x+y and U^b=x^2 + y^2 and
endowments are A(2,1) and B(1,2),
two allocations are given,
allocation 1-A(1.5,0) and B(1.5,3) and
allocation 2-A(1.5,1.5) and B(1.5,1.5).
Allocation 2 is not Pareto optimal because there exist a feasible allocation such as A(0,3) and B(3,0) which improves the welfare of B without affecting the welfare of A.
Allocation 1 is Pareto optimal because there is no feasible way to improve on it.

[Sumit]

Sir I hv two more doubt...
1) we can't take initial endowment as a pareto efficiency coz there is a feasible solution to improve welfare of B without hurting A's welfare (i.e A(1,2) & B(0,3)...right???
2)suppose If there is third Allocation in the question as follow: A(0,0) and B(3,3)..will it also consider as a pareto efficiency????

[Amit Goyal]

Please read the definition of efficiency carefully and you would know. This is what efficiency means:
We say that a feasible allocation A is NOT efficient if there exist some feasible allocation B that can improve the welfare of at least one of the individuals of the society without reducing the welfare of the others i.e. u_i(B) > u_i(A) for some i and u_i(B) >= u_i(A) for all i.
We say that a feasible allocation A is efficient if there is no way to improve on it i.e. it is NOT NOT efficient.

[Sumit]

got it sir... thanks a ton...

[maahi]

How do we know that we indeed have a pareto efficient allocation - by hit and trial ????