Pareto optimality in concave preferences

[Asd1995]

Suppose 2 individuals have the utility functions x^2 plus y^2, where x (total)=y(total)=1

Find pareto optimal locations.

[Algaeconomics]

I haven't given it a hard thought but is it all the edges of the box?

[Asd1995]

Looks like it is. Is there a non-edgeworth box way to go about it? Like, if there were 3 such agents with same preferences?

[Algaeconomics]

I dont know :|

BTW the first question in PEB, how did you solve it?
 
Did you make a straight line x1+x2=100 and then used hit and trial to observe that if Aditya has x between 20 and 70 ( or Gaurav has x between 30 and 80) then he cannot be made better off hence the first allocation is pareto efficient?

[Seecha]

alibaba ...plz help me out me with same paper 's PEB 5th , 20th and 21st .. please

[Algaeconomics]

20th
first take a two people economy and see the optimum, since it is unregulated we know the optimal for each person is v'(x)=0 since the function v is same for all they will maximize at the same x. ( given it is inverted U) so now even if the economy has more population say 3 people the answer will be the same regardless that all of the three operate at that point where v(xi) is maximized.

now the question is asking km driven per person
in first case it was 2x/2= x
and second case 3x/3=x

so it's same.

Others I did not attempt for the exam. I'll try and let you know.


asd1995 did you try these??

[Asd1995]

No, I wrote the utility of both as a function of Aditya's work hours, and graphed both functions (its basic graphing idk if eco students are familiar with it). Identified both locations, and from there it was easy to see if I could keep one's utility constant and increase the other's. You'll get it. In case you don't, I'll graph it out for you and put up a picture.

[Asd1995]

First one just reduces to each person maximizing his own utility without any constraint. So it won't change.

Did not attempt the tax one.

[Algaeconomics]

So is it like two parabolas open downwards with centre at (20,0) and (70,0) and then we compare points for pareto optimality?

[Asd1995]

Not parabolas, each graph is a pair of straight lines with a kink at the point where it touches the x axis. So like a parabola but both "arms" of the parabola are straight lines.

[Algaeconomics]

yaar I just solved the 5th question
It was silly :| could have solved it then too.

so the condition is

The utility obtained without taking  effort < utility obatined with taking effort

p(0)U(W-T) + {1-p(a)}U(W)  <  p(a)U(W-T) + (1-p(a))(U(W)) -Aa^2

Solving for A you can find the value of A for which effort will be undertaken

[RichaS]

Shouldn't it be the bottom and right edge of the box instead of all edges?

[Algaeconomics]

It was an equation of parabola and not mod fn though the logic remains the same. thanks

[Asd1995]

Sorry I confused it with another one I solved recently. Apologies. Of course the answer wouldn't change.

[Asd1995]

For 5th you could just observe that for a=0 he will never undertake any effort, so it has to be a root of the equation. Similarly the other factor should vanish there.

[Asd1995]

You could use symmetry arguments to see that all 4 sides are equivalent. I could interchange x and y in the problem and nothing would change.

[Algaeconomics]

If we fix the utility of the second person such that the end points of concave fn lie on the upper and right edge then max utility of the first person will be at the corner points of the first?

Im not sure correct me if I'm wrong please!

[Seecha]

alibaba .. i think you got this equation in 5th after doing bit of shifting from LHS to RHS ...
i have done it myself ... i got the ans. thanx for hint . :)

[Algaeconomics]

Yeah was a bit lazy to write the whole thing down  

[Algaeconomics]

Yeah was a bit lazy to write the whole thing down