Doubt!!

[Sumit]

Guys help me in these three question...n plz show proper working..







In last question I'm getting correct answer If I do it by plotting budget line n indifference curve...but as I try to do it mathematically i.e without help of Budget line n IC ...I'm getting wrong answer...so plz show latter one..

Thanks in advance...

[Devika]

Is k= 3/2?

[Sinistral]

3.53
plot the domain. plot x+y>1/2. overlapping region is our area of interest:
P(X+Y>1/2)= Integral (y=1/2 to 1) Integral (x=0 to y) f(x,y) dx dy + Integral (x = 0 to 1/4) Integral (y=1/2 -x to 1/2) f(x,y) dy dx +  Integral (x = 1/4 to 1/2) Integral (y=x to 1/2) f(x,y) dy dx

Do you want to match your answer? because I haven't  solved the above integral yet.

3.49
Given: 0<x<1
Given: -x<y<x ==> y <|x|

since x is always positive we will take that portion of the graph y<|x| which is in first quadrant.
so, Integral (x= 0 to 1) Integral (y=0 to x) kx(x-y) dy dx  =1
                                                                            k= 8

[Devika]

Damn, x_x

I did y < |x| => -x<y<x => -1 <y <1. Forgot about the constraint given by the pdf!

Also sinistral, where are you doing prob distributions from? I've noticed you are crystal clear with distribution and I marvel at the lucidity of your explanations. Which book are you referring to?

[duck]

Hey.. :)

40) As px/py=1/2, consumer would always like to consume x and zero units of y.
One way to look at it is graphical way. Another rough way could be:
If x<y then, U=2x. So, he would always like to consume x.
If x>y then, U=x+y. As px<py, so again would like to consume only x.
So, in order to maximise his utility he would always like to consume only x.

[Sinistral]

hey,
I will be writing DSE entrance with Option B. However, not a day goes by when I don't doubt my decision. Prob & Stats and Real Analysis for that matter, are the core of option B syllabus. So I need to be well versed with Statistics.

[Sumit]

hey Sinstral,
Answer given in book is 3.53=1-1/2ln(2)=0.6534.
And for 3.49=K=2...I think your answer K=8 is correct....but I didn't get how you come up with this Integrals in 3.53...Could you show graph for it....

@thanks duck..:-D

[ankur]

@duck
Q40..
It is perfect complement function
So 2x=x+y
x=y
Optimal soln(300/100+50,300/100+50)..
(2,2)
By ur soln the utility is zeo

[Mauli]

hi ankur:)
(6,0) means x and y are 6 and 0 respectively.
the utility function given us min(2x,x+y) =(2*6, 6+0)
utility is 6.
and as duck explained above the equilibrium cannot be (2,2).

[duck]

Hey.. :)

Answer for 3.49 will be "2" only.
Please find attached the solution.
Stats.png
You can recheck: If k=8 then, it will not serve as a pdf.

[duck]

3.53) Attached is the graph.
3.53.png

[Sinistral]

how stupid of me!! :(
-x<y<x => |y|<x (and NOT y<|x| as I did earlier) . So the required graph will be the one drawn by duck and accordingly the value of k will change.

thank a lot Duck for pointing that out.