ISI INTERVIEW PREP

[anon_econ]

Hi Preet. Thanks for ur lovely words. I really enjoy discussing problems on this forum :-)
I'm really not sure about this question. I was considering not just how much money I have and how much the other person is likely to have, but also how much money the other person is expected to have if he too chooses to swap. And deepak if u r using the logic of the two envelopes problem, the answer would be 1, right?

[anon_econ]

also i'm not quite sure how this is analogous to the two envelopes problem. there u were given that one envelope has twice as much money as the other. what about here?

[deepak]

Vasudha, you could reinterpret the 2 envelopes problem as not being twice as much, but the 2 envelopes having different sums of money. Therefore, one of them must have more money than the other, similar to our case, where one person would have more than the other. So it reduces to the same problem.

And if this were the case, they wouldn't swap right? The expected value in the case of either envelope is the same, and there is no point in swapping.

Though I'm not completely convinced by my own argument :P

[deepak]

Ok I floated this problem to a few people I know and they all seem to concur on 1/4, the simple answer one gets by doing the following :
P1 = P(1st guy gets less than the expected value) = 1/2
P2 = P(2nd gets less than the expected value) = 1/2
P(swap) = P1 * P2 = 1/4.
If one seeks to prove this without going the expectation way, it can still be shown that the probability is 1/4.

One more, elegant proof is this :
Each player has a strategy of swapping S or retaining R. So the different combinations of decisions are
RR,SR,RS and SS. A swap occurs only in the SS case and the probability of this happening is 1/4

[anon_econ]

Umm but deepak each player knows that the other guy would also swap only if he gets less than the expected value. y wont he keep that in mind?

[Mr. Nobody]

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[Mr. Nobody]

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[deepak]

Well like if Ram says 1/4 is not in the list of options then I'm truly stumped. Either we are missing something that crucially defines the question or we are still finding a formal proof hard to come by.

[deepak]

Since 1 unit of labor produces 1 unit of X and similarly Y, the 2 production functions are
X = L1 and Y = L2 and it is given that L1 + l2 = 10
Now the marginal cost of X or Y wil ljust be the labor wage w. This is also the price. ie Px = Py = w
Also, U = root(xy) and income m = 100
We also found out now that Px = Py = w.
So Xd = 100/2w = 50/w
Yd also = 50/w

But L1 = 50/w as is L2 = 50/w.
L1 + l2 = 10 => 100/w = 10 so w = 10

[Mr. Nobody]

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[deepak]

Ok I think I can prove myself wrong :/

I assume independence in the step where I multiply the 2 probabilities ie P(Swap) = P(P1^P2) = P1 * P2

But... I think assuming independence here is wrong. Let's say the 2 events are
E1: Person 1 swaps if he thinks he has a lower number than the other guy ie Y>X
E2: Person 2 swaps if he thinks the same ie X>Y

So, for the 2 events to be independent,
P(E1|E2) = P(E1)
ie P(Y>X|X>Y) = P(Y>X)
but, if it is given that X>Y, then we knwo that P(Y>X|X>Y) is 0, thereby proving the assumption of independence wrong.

So now, how does 1 arrive at the right answer? formally :P

[n.saish]

Suppose n numbers are drawn uniformly from [0, a], (a > 0).
Show that the expected value of the second highest number is a*(n-1)/(n+1)

Am getting [(a^2)*(n-1)]/[n*(n+1)]....

[neha]

@saish ,, i am getting a*(n-1)/(n+1),,, First of all u should get the PDF of the second highest value (z) ,, which is [n(1-n)*z^(n-1)/a^n ] + [n(n-1)*z^(n-2)/a^(n-1)] ,, now the range of z is 0 to a ,, we can integrate

[neha]

This is regarding the question number 6 of DSE 2004
http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/26_jun_2004_option_a.pdf

why the answer is option (b)? Is nt it about favoring the individual 1???  I mean the set of allocations is pareto optimum,, and hence can be competitive under certain conditions but this set does not contain all the competitive allocations.

[Rain Man]

@ Neha ..

how did u get the pdf for the second highest number (z) ?

please elaborate..

thanks

[neha]

@ rain man

the pdf is similar to the one discussed in the following link
http://discussion-forum.2150183.n2.nabble.com/file/n7578969/probIIhi.png;cid=1341502738193-855

If u find the cumulative distribution function of z and then differentiate it, you will get the PDF of z.

Hope it helps.

[Saish]

I'm not getting the pdf you are...
Pls.correct me wherever I'm wrong...
See the way I'm thinking is that the probability of z being the second highest number is the probability of   n-2 numbers being smaller than z and one greater than z
So the pdf = (n-1)[(x/a)^(n-2)]*[(a-x)/a]
... the (n-1) term has come as there are n-1 different numbers that can be greater than z...
There difference in your pdf and mine is a term (n/a) is missing... which is also the problem with the answer I've got.... could you tell me where the mistake is?

[neha]

Saish, your mistake is while deriving the expression for CDF, you are missing out the case when all xi 's will be smaller than z.

You can refer to the  link
 http://discussion-forum.2150183.n2.nabble.com/file/n7578969/probIIhi.png;cid=1341502738193-855

[Preet]

Neha, the only way the competetive equilibrium can exist is if good y is a free good. Now, if good y is a free good then Agent 2's income will be zero. hence he cannot consume any of the good x. Agent 1 will consume all x=999 and y belonging to (999,1000).. Hope this makes it clear.

neha wrote

This is regarding the question number 6 of DSE 2004
http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/26_jun_2004_option_a.pdf

why the answer is option (b)? Is nt it about favoring the individual 1???  I mean the set of allocations is pareto optimum,, and hence can be competitive under certain conditions but this set does not contain all the competitive allocations.

[neha]

preet thanks so much for replying,

tell me if i am wrong in my reasoning,, price of x cant be zero 'cause in that case agent 2 will demand 1000 units of y which are not available, hence px =0 and py=1 cant be a set of competitive prices.

Also can you tell me q 47 of dse 2011
http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/2011-option-a.pdf