dse 2012 q45 please help

[kam]

which set is convex and why, please help

[MR]

U is. If you draw the sets, or even by just basic looking at them and knowledge of the definition is enough.

[maahi]

its been long sice i read d defn can u state it here ?

[MR]

Hey Maahi,
See informally it is like if you draw a set or any figure and try to join two points and draw a line to do so, the line should be within the given figure (set). Then the figure or set is convex or if the line is even a bit out of the set it is concave.Formally http://en.wikipedia.org/wiki/Convex_set. As even I don't remember the formal one word to word. :| Hope this helps. =))

[maahi]

what is dis c here ? f(x,y) < =c

[MR]

C?

[MR]

Ok ok c, I guess you can take any number which belongs to R++. Just the way you can take any number as x y from R++. I am no pro in maths. This is how I deal with these questions i.e. in layman terms. So any pro maths point of view will be really appreciated.  =))

[maahi]

lets wait till brilliant people come and ans .

[MR]

True. I have real hard time deciphering all the a b c d's of the paper. :(

[duck]

Hi.. :)

"c" is some strictly positive real numbers.
Interpret "U" as the upper contour set i.e. set of all those points (x,y) which gives the functional value greater than or equal to "c."
Interpret "L" as the lower contour set i.e set of all those points (x,y) which gives the functional value lesser or equal to "c".
Interpret "I" as the indifferent set i.e. set of all those points (x,y) which gives exactly the same functional value of "c". (Eg: Indifference curve, Isoquants).

In this particular question, we've:
f(x,y)=x+y+xy and suppose, c=10.
Now, U= {(x,y) | f(x,y) ≥ 10}   or {(x,y)|x+y+xy ≥10}
Similarly, figure out L and I.

One way: Plot the level curves and then check whether any of the U, L or I is convex or not.
Please find attached its graph.
Dse.45.png

In the graph, you can see only "U" is a convex set as any two points taken in U, the convex combination also belongs to U.

[duck]

Another way: Use bordered Hessians to check whether a function is quasiconcave or quasiconvex.
[If a function is quasiconcave then, upper contour set "U" is a convex set]
[If a function is quasiconvex then, lower contour set "L" is a convex set]

[maahi]

Hi


 thanks sab yaad aa gaya

 but how did u plot d graph its 3d no ?

 also if you can respond to these questions . i know its too much to ask for
http://discussion-forum.2150183.n2.nabble.com/DSE-questions-tp7582345.html;cid=1371461030116-872

 

[duck]

I have plotted the level curves.

[maahi]

ya got it thanks

[HRC ROCKZ]

if we use bordered hessians,the test fails n v cant tell whether its concave or convex or even quasiconcave or quasiconvex.

[Ankit]

Google search for "plot x+y+x*y". You will get a nice plot. You can visualize it well!

[mrittik]

but DUCK then 3 curves are going to be Convex....how do you find only opt b is correct?

[duck]

Hi Mrittik,

The question is "which set is a convex set".

[Akshay Jain]

Welcome back duck...