GENERAL EQM and prob doubts

[k]

 
Q1)  Xavier and Yvette are the only two persons on a desert island. There are only two goods, nuts and berries. Xavier's utility function isU(Nx, Bx) = Nx .Bx.  Yvette's utility function is U(Ny, By) = 2Ny + By . Xavier is endowed with 3 units of berries and 8 units of nuts.Yvette is endowed with 6 units of berries and 8 units of nuts. In a competitive equilibrium for this economy, how many units of berries does Xavier consume?
a)      12.50
b)      19
c)      7.50
d)      9.50
e)      None of the above
 
 
 q2) Use Baye's Theorem to solve this problem, A paper bag contains two red balls and one blue ball. A plastic bag contains three blue balls and one red ball. A coin is tossed. If it falls heads up, the paper bag is selected and a ball is drawn. If the coin falls tails up, the plastic bag is selected and a ball is drawn. If a red ball is selected, what is the probability that it came from the paper bag   _______
A)  8/11
B)   1/3
C)   1/8
D)   3/8
 
 

Question 3.
Find the probability that of 25 randomly selected students, no two share the same birthday. _______
A) 0.431
B)  0.995
C)  0.569
D)  0.068
 
q 4)
Find the probability that of 25 randomly selected students, at least two share the same birthday. _______
A) 0.569
B)  0.068
C)  0.432
D)  0.995

[Prerna Rakheja]

Hi Preety

Ques 1) d
Ques 2) a
Ques 3) 365*364*363*...341/(365)^25 should be approximately 0.431
Ques 4) 1 - 0.431 = 0.569

[MR]

Hey Prerna,
How did you get d for the first question? Can you show the working?

[Sumit]

Hi prena,
Your first question answer is wrong....How could he consume 9.5 berries..If the total berries available is 9...
answer will be either c) or e)...

[k]

hey prerna how u have done calculation of 3 ques?

[Xiang]

how to solve 1st question? procedure?

[Sumit]

Answer for q1. is e) None of the above.

the easiest way to do it by using options....
First of all the max. berries Xavier can consume is 9...
So,the option a),b) & d) eliminates
Now, we have to check whether option c)7.5 is correct or not, If it is wrong the answer will be e)none of the above..

How to check for c)7.5..
Xavier demand for berries (Bx)=Mx/2pB=(8pN+3pB)/2pB=7.5..
By Solving above you will get PN/pB=(3,2).
Now, all we have to do is to check at the prices (3,2) whether market clearing or not...

At (3,2) Xavier's Demand: (Nx,Bx)=(5,7.5)
Yvette's utility fn U(Ny, By) = 2Ny + By.
Yvette's income My=8PN+6pB=36
Yvette Budget constraint 3Ny+2By=36...

Now, Yvette Max its utility at Ny=12 and By=0...
We can see that mkt is not clearing at price(3,2)(bcoz for any of the good supply is not equal to demand)...so, this is not a competitive prices...Thats why answer is e).

[MR]

Hey Sumit,
Thanks for the answer. I was looking into the question in the following manner:
That say price of nuts be 1, and price of berries be p. So Xavier's endowment is worth 3+8p and Yvette's endowment is worth 6+8p. Now Xavier's demand for nuts will be 3+8p/2 and for berries will be 3+8p/2p. And Yvette consumes 6+8p of nuts when p>1 and consumes 6+8p/p of berries when 1>p and another case for p+1. And I was trying to equate both their demands for a particular commodity at a price case. I was getting that 1>p but could not arrive at the exact consumption. Can you tell why or what went wrong? And when to use this method or not. Will be very helpful. :)

[Akshay Jain]

if we follow the normal method to calculate the competitive equibrm den we will get amt of berries for x - 9.5 but becoz the total amt available is 9 the ans is none of d above..

[Sumit]

First of all you doing the same mistake as Prerna did...Xavier endowment for berries is 3 not 8..I think thats why you are not getting the right answer...Also the method of assuming one good price is 1 and other P..also correct.

[MR]

Thanks Sumit. :)

[Do bats eat cats?]

Has anyone actually solved the first question completely? I would really appreciate it if someone could confirm my approach.

Let nuts be the numeraire good, the price of berries will be p.
So, the value of Xavier's endowment is 8 + 3p and the value of Yvette's endowment is 8 + 6p.
I found Xavier's optimal demand for berries, which came to (8 + 3p)/2p.

Then I considered three cases (this is the bit I'm confused about)

1. p > 0.5
In this case Yvette will not consume any berries, so By=0 and Bx=(8 + 3p)/2p.
Bx + By = 9.
So, (8 + 3p)/2p = 9 or, p = 8/15 and Bx = 9, which is the required answer.

2. p = 0.5
Bx = (8+3(0.5))/2(0.5) = 9.5, which is not possible. So, we ignore this case.

3. p < 0.5
Here, Ny = 0, and By = (8 + 6p)/p
Again, Bx + By = 9.
So, (8 + 3p)/2p + (8 + 6p)/p = 9 or, p = 8, which is contradictory. So, we ignore this case as well.

[Prerna Rakheja]

Hi All,

Apologies for the mistake, it should be e. Total endowment of berries is 9 hence d cannot be the case.