Strong vs Weak Pareto Efficiency - May 23

[Amit Goyal]

May23_StrongVSWeakPE.png

[AJ]

I got,

Strongly Pareto Efficient Allocations:
      (x1,y1)  (x2,y2)
      (0,2)        (1,0)
      (1,0)        (0,2)

Weakly Pareto Efficient Allocations:

All allocations where either x1 or y1 is discrete!

Example: (0,y1)  (1, 2-y1)
             (1,y1)  (0, 2-y1)
             (x1,0)  (1-x1, 2)
             (x1,1)  (1-x1, 1)
             (x1,2)  (1-x1, 0)

[AJ]

@Sir,
Please reply if its wrong...

[anon_econ]

the set of strongly efficient allocations is the same as what AJ got, and the set of weakly efficient allocations consists of all the points that i've highlighted in red (including the end points). i'm not sure about this!

[aditi5000]

Stong pareto allocations - same as AJ
Weak pareto allocations - (a,0) and (1-a,2) ; (1,a) and (0,2-a) where a belongs to (0,1)

[Amit Goyal]

AJ, Aditi and Vasudha, All of you got strongly Pareto efficient allocations correct. But think about weakly pareto efficient allocations again. For example:- check that ((0.5, 1), (0.5, 1)) is weakly efficient but is not included in the set you illustrated.

[AJ]

@Amit Sir,

It is included in the set I wrote...
.. "all those allocations where either X1 or Y1 is discrete"....

Can u give  an allocation, where both goods are not discrete and still its weakly pareto efficient..
So, that I can compare it with my answer..and find the mistake...

[aditi5000]

Sir a weak pareto efficient allocation is one that makes only one agent better off naa? (without making both better off..) So can't any allocation like (a,b) and (1-a, 2-b) for a,b belonging to (0,1) would be weakly PO? Since their endowments haven't been mentioned..

[anon_econ]

what about this?

[Amit Goyal]

Thats right Vasudha. Well done.
Let F denotes set of all feasible allocations (set of all allocations in the edgeworth box)
Let x(i) denotes i's consumption of good x, and y(i) denotes i's consumption of good y
Set of weakly efficient allocations is given by
{((x(1),y(1)),(x(2),y(2))) ∈ F| y(1) = 2} U
{((x(1),y(1)),(x(2),y(2))) ∈ F| y(2) = 2} U
{((x(1),y(1)),(x(2),y(2))) ∈ F| max (x(1),y(1)) = 1} U
{((x(1),y(1)),(x(2),y(2))) ∈ F| max (x(2),y(2)) = 1}

[Amit Goyal]

Hi AJ Thats not the precise description of weakly efficient set. Consider an allocation
((1,1.5), (0,0.5)) This is part of the set you listed but is not weakly efficient as ((0,2), (1,0)) beats it.

[aditi5000]

Sir I think my GE concepts are terrible. Please recommend where to study it from. First time round, I studied it from HRV. But I need more practice on such questions. I keep getting confused...

[AJ]

okay.. This means I also took lower left and upper right vertical lines of vasudha's edgeworth box too..which are infact not weakly pareto optimal..!! :(

@ AMIT sir, vasudha
what is the best way to proceed in G.E. questions..??

like for competitive Equilibrium.. I start with demand functions of both individuals and then take one good.. and equal the demand with endowment of that good.. (and normalize one of the price)

And for Pareto optimal.. I take some random points, see their behavior.. and draw conclusion on that basis... but this way ofcourse I can make mistakes like above...
So, is it always better to try to find points on edgeworth box...like vasudha did.. and then draw conclusion...

Vasudha, how u proceed for these questions...??

[Amit Goyal]

Hi Aditi,

A very good reference for this topic is
Ch-15 of Microeconomic Theory by Mascollel, Whinston and Green

But it would be better if you do Ch 1, 2 and 3 of the same before doing chapter 15. The book is rigorous, and extremely clear in its presentation.

If you want slightly less rigorous treatment, Ch - 17 of Microeconomic Analysis by Varian could be a reasonable choice.

[Amit Goyal]

AJ, The "best" way varies with the problem. The way you listed is good and will work for all the questions provided you are careful while doing computations but there are usually multiple ways to solve the same problem. And the best is learned through experimentation and practice.

[AJ]

@Sir,
Okay, I will see the above mentioned book too..
Sir it will be really helpful if you can see my following solution for GE from 2006 DSE paper..
Please.. And tell me if there is any mistake in my method or "Concepts" most importantly...
I know I am bugging U a lot, but I haven't studied GE in my course as such and I always make mistake in 1st go.. I want to attempt those 3 questions right which we always get in DSE.