DSE 2013 Doubts

[ViV]

The next TWO questions are based on the following model:
Suppose that there are two goods, which are imperfect substitutes of each other. Let p1, p2
denote the price of good 1 and good 2, respectively. Demand of good 1 and
good 2 are as follows
D1(p1, p2) = a - p1 + bp2; D2(p1; p2) = a - p2 + bp1
where a > 0 and 1 > b > 0. Both of the goods can be produced at cost c per
unit.
QUESTION 7. Find the equilibrium prices, when good 1 and good 2 are
produced by two different monopolists.
(a) p1 = p2 = a+c/2-b

(b) p1 = p2 = a+c/1-b

(c) p1 = a+c/2-b, p2 = a+c/1-b

(d) p1 = a+c/1-b,  p2 = a+c/2-b

QUESTION 8. Find the equilibrium prices, when both the goods are pro-
duced by single monopolist.
(a) p1 = p2 = a+c-bc/2-b

(b) p1 = p2 = a+c-bc/1-b

(c) p1 = p2 = a+c-bc/2(1-b)

(d) p1 = p2 = a+c-bc/2



Suppose three players, 1, 2 and 3, use the following procedure to allocate 9 indivisible
coins. Player 1 proposes an allocation (x1; x2; x3) where xi is the number of
coins given to player i. Players 2 and 3 vote on the proposal, saying either
Y (Yes) or N (No). If there are two Y votes, then the proposed allocation
is implemented. If there are two N votes, the proposal is rejected. If there
is one Y vote and one N vote, then player 1 gets to vote Y or N. Now, the
proposal is accepted if there are two Y votes and rejected if there are two N
votes.
If 1's proposal is rejected, then 2 makes a proposal. Now, only 3 votes
Y or N. If 3 votes Y, then 2's proposal is accepted. If 3 votes N, then the
proposal is rejected and the allocation (3; 3; 3) is implemented.
Assume that, if the expected allocation to be received by a particular
player by voting Y or N is identical, then the player votes N.

QUESTION 28. If 1's proposal is rejected and 2 gets to make a proposal,
her proposal will be
(a) (0; 5; 4)
(b) (0; 4; 5)
(c) (0; 6; 3)
(d) (0; 3; 6)
QUESTION 29. 1's proposal will be
(a) (5; 0; 4)
(b) (4; 0; 5)
(c) (3; 6; 0)
(d) (6; 3; 0)
Just solve 29 only.

QUESTION 42. A bowl contains 5 chips, 3 marked $1 and 2 marked $4.
A player draws 2 chips at random and is paid the sum of the values of the
chips. The player's expected gain (in $) is
(a) less than 2
(b) 3
(c) above 3 and less than 4
(d) above 4 and less than 5

QUESTION 44. A certain club consists of 5 men and 5 women. A 5-
member committee consisting of 2 men and 3 women has to be constituted.
Also, suppose that Mrs. F refuses to work with Mr. M. How many ways are there of constituting a 5-member committee that ensures that both of them
do not work together?
(a) 50
(b) 76
(c) 108
(d) None of the above

QUESTION 45. Suppose, you are an editor of a magazine. Everyday you
get two letters from your correspondents. Each letter is as likely to be from
a male as from a female correspondent. The letters are delivered by a post-
man, who brings one letter at a time. Moreover, he has a `ladies rst' policy;
he delivers letter from a female rst, if there is such a letter. Suppose you
have already received the rst letter for today and it is from a female corre-
spondent. What is the probability that the second letter will also be from a
female?
(a) 1/2
(b) 1/4
(c) 1/3
(d) 2/3

QUESTION 46. On an average, a waiter gets no tip from two of his cus-
tomers on Saturdays. What is the probability that on next Saturday, he will
get no tip from three of his customers?
(a) (9/2)e^-3
(b) 2e^-3
(c) (4/3)e^-2
(d) 3e^-2

[ViV]

Please show me your workings also because most them I have already tried....

[mrittik]

q46. its just the poisson distribution.

the newspaper problem is the same fav prob of DSE...its same as boy-girl probability...but in different guise..its a discreet probability distribution problem.....you may find in 2004 DSE boy-girl problem

[kangkan]

@ viv..refer to this thread..you will find most of your ansewers here

[kangkan]

http://discussion-forum.2150183.n2.nabble.com/DSE-2013-Paper-Discussion-td7584703.html

[ViV]

Thanks Kangkan for your response but the questions I have posted were not discussed there...

[kangkan]

alright...i will post by the evening :)

[phelps.phan]

Q 28: Given player 1's proposal is rejected. 2 will propose such that he is better off than (3,3,3) and 3 will accept if its better than (3,3,3).
Option d is ruled out. for 2 it is not better off than (3,3,3) while he has an advantage of proposing.
Also he would not choose Option c since player 3 might reject it since player 3 can still get the same utility from (3,3,3) after rejecting.
Player 3 would accept any of option a and b. So player 2 will try to maximize his utility selecting a.

Q 29: Option d: ruled out as player 2 can reject it without any loss and player 3 will definitely reject it.
Option c: might be accepted but he gets a utility of 3.
Option a: its rejected by player 2. It can be rejected by player 3 too as we know if player 1's proposal is rejected, he would accept the proposal (0,5,4). Player 3's utility is same as of (5,0,4). So player 1 would not take the risk for player 3 to reject his proposal.
If he proposes (4,0,5) player 3 cannot reject it as he is better off than (0,5,4).
Also player 1 would prefer (4,0,5) to (3,6,0). So answer b.

[Akshay Jain]

On an avg the waiter doesnt get tip from two of his customers so it follows a poission distn with lambda=2
the prob that on any day he will not get tip from 3 of his customers is (e^-2)*2^3/3!
=(4/3)*e^-2

[Akshay Jain]

prob that a letter is from male=1/2=prob that it is from a female
P(event A)=prob that both letters are from female=1/2*1/2=1/4
P(event B)=prob that 1st letter is from female and 2nd is from a male=1/2*1/2=1/4
P(event C)=prob that 1st letter is from male and 2nd is from a female=1/2*1/2=1/4
but the above event C shud be included in event B because no matter in what order the postman gets the letter he will deliver female letter 1st whenever he has a letter from a female
given that 1st letter is from a female
prob that 2nd letter is also from a female is (1/4)/[(1/4)+(1/2)]=1/3

[kangkan]

7 and 8....

Part 1.
q1=a-p1-bp2     q2=a-pe-bp1

Now eliminate p2 and express p1 interms of q1 and q2

p1= a/(1-b)  -q1/1-b^2   bq2/1-B^2

Now since the two firms price independently,it is a a case of cournot equilibrium...let q2=q2* (expected) and solve for the cournot equilibrium....after goin thru a lot of algebric simplification ,we wud get p1=p2= a+c -B^2/( [(2+b)(1-b)]..now since b is quite small,ignore the sqaures of b p1 simplifies to a+c/@-b

parrt 2..if the same...just solve the maximization problem with q1 and q2 as the control variables.

I wud have loved to put a pic,but my phone has no camera

I see that the other problems have already been solved 

[ViV]

Thank u soooo much every1...:-)

[rongmon]

Having eliminated p2 and expressed p1 in terms of q1 and q2, how does one proceed from there?

[Pragya-2]

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The author has deleted this message.

[Dreyfus]

@kangkan....7 and 8 is a price setting situation not quantity setting....den y r u using profit function for quatuty setting?

[kangkan]

hi,...i just used the nash equi condition..given an expectation of p2 ,or implicitly an expectation of q2, how wud i maximize my profits....

[kangkan]

hi...thanks for pointing it out...i just realized my approach was wrong...and arrived at a much simpler solution..thanks

[rongmon]

Kangkan, could you share the workings for the price setting problem, i've hit a dead end.

[kangkan]

Hi...let P(e) denote the expected prices

q1= a-p1+p2(e)*b
=> p1=a-q1+P2(e)*b

the profit maxi prob is

Max q1( a-q1- p2(e)*b)) - q1*c

Solving we get q1= a+bP2(e)-c/2 and a similar one for q2

Now remeber this equation must satisfy the demand equations in equlb

So a-c+bp2(e)/2= a-p1+bP2(e)

by the symmetry of the problem we can say that p1=p2 =P*(say)

Subsituting in the previous equation we get p*=a+c/2-b  :)

[rongmon]

Thanks Kangkan!