jnu mphil questions..

[shelly gupta]

Q1. Suppose in a set of 4 alternatives {x, y z, u4, x and y are Pareto-optimal while z and w
are not. From this we can infer that
(a) x is Pareto-superior to z
(b) x is Pareto-superior to w
(c) x is Pareto-superior to both z and w
(d) None of the above



Q2. For the next two questions,, consider the following :

There are two individuals, 1 and 2. Each individual has an initial endowment of 30. There is a
machine with the following property : Should individuals 1 and 2 provide respectively
endowments xi and x2 to the machine, the machine first computes the aggregate contribution,
xi + x2. This done, the machine responds by providing each of the individuals fresh
endowments equal to 5(x1 + x2)1/2. Thus the utility of individual i from the contribution profile
(x1, x2) is Ui (x1, x2) = 5(x1 + x2)1/2 + (30 - xi ). Note also that the endowment given to the
machine by individual i, xi, cannot exceed her initial endowment of 30.


Q.  Which of the following contribution profiles (x1, x2) maximizes the sum of the utilities of
the two individuals, U1(x1, x2)+U2(x1, x2)?
(a) (30, 30)
(b) (15, 15)
(c) (0, 25)
(d) None of the above

Q.  Suppose the individuals make their respective contributions, xl and x2,
simultaneously. This means that when an individual chooses her contribution level, she
is unaware of the contribution level chosen by the other person. For this
simultaneous-move game, which of the following contribution profiles constitute a Nash
equilibrium?
(a) (15, 15)
(b) (0, 25)
(c) (25/8, 25/8)
(d) None of the above

[Granpa Simpson]

Q1. c...this is because if x is pareto optimal, it means no other options available is pareto superior given the set of pareto efficient frontier, so c).
Q2. (30,30)..LPP can be used..!!!

[vandita24x7]

Subhayu fr q.2 u see utility is more maximized when (15,15) is put and even further when (0,25) is put. How did u get a definite n sure answer using LPP ???

vandita

On 26 Apr 2014 10:02, "subhayu [via Discussion forum]" <[hidden email]> wrote:

Q1. c...this is because if x is pareto optimal, it means no other options available is pareto superior given the set of pareto efficient frontier, so c).
Q2. (30,30)..LPP can be used..!!!

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[Granpa Simpson]

@vandita: if you put (15,15) the sum of the utilities will be 180, and if you put (25,0) or (0,25) it will count to 160 too but if you put (30,30) the sum of the utilities will be 300..!!!..ultimate goal is to maximize utility so that bundle will be taken for which utility will be maximized..LPP may or may not be used..even if u use LPP it will give same results..isnt it so..??

[vandita24x7]

Hi subhayu, the utility function is 5(x1+x2)^1/2 + (30-xi)

If u hv used this... pls show me ur solving method.

vandita

On 26 Apr 2014 10:21, "subhayu [via Discussion forum]" <[hidden email]> wrote:

@vandita: if you put (15,15) the sum of the utilities will be 160, and if you put (25,0) or (0,25) it will count to 160 too but if you put (30,30) the sum of the utilities will be 300..!!!

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[Granpa Simpson]

Hello vandita: Even I have used the same utility function, in the question its given that u will have to find out x1 & x2 s.t the sum of utilities is maximized which means the objective function will be z=[5*(x1+x2)*(1/2)+(30-x1)]+[5*(x1+x2)*(1/2)+(30-x2)]...!!!..given x1<=30, x2<=30...!!!

[vandita24x7]

Yess yess i used this objective functn only... only that 5(x1+x2)^1/2 not *1/2.
Anyway show me ur solving... perhaps m making a silly error and cant rectify :-(

vandita

On 26 Apr 2014 10:35, "subhayu [via Discussion forum]" <[hidden email]> wrote:

Hello vandita: Even I have used the same utility function, in the question its given that u will have to find out x1 & x2 s.t the sum of utilities is maximized which means the objective function will be z=[5*(x1+x2)*(1/2)+(30-x1)]+[5*(x1+x2)*(1/2)+(30-x2)]...!!!

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[Granpa Simpson]

I multiplied (1/2) and u took it in power..so the answer is differing i guess..nywa i just maximized z=[5*(x1+x2)*(1/2)+(30-x1)]+[5*(x1+x2)*(1/2)+(30-x2)], s.t x1<=30, x2<=30, x1>=0,x2>=0...u will get four points (0,30), (30,0) (0,0) and (30,30) out of which z will be maximized at (30,30) with a value of 300...!!!!

[vandita24x7]

Ohhh subhayu.. thats what i was telling u. The utility fn has 5* sqrt(x1+x2) . This way (30,30) is not max.

Could solve the 3rd/last questn?? Its the 2nd part of this utility questn.

vandita

On 26 Apr 2014 10:47, "subhayu [via Discussion forum]" <[hidden email]> wrote:

may b I am wrong..because I multiplied (1/2) and u took it in power..so the answer is differing i guess..nywa i just maximized z=[5*(x1+x2)*(1/2)+(30-x1)]+[5*(x1+x2)*(1/2)+(30-x2)], s.t x1<=30, x2<=30, x1>=0,x2>=0...u will get four points (0,30), (30,0) (0,0) and (30,30) out of which z will be maximized at (30,30) with a value of 300...!!!!

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[Granpa Simpson]

Vandita: Yes ur right if there is a square root then LPP cannot be used...did u check the question..??trying the second part..!!

[Granpa Simpson]

I think for Q3 d will be the answer because in all of the cases the utility of both the agents can be increased if they move to (30,30)...!!!!(N.B: I am using the utility function with (1/2) multiplied...)

[vandita24x7]

Exactly lpp cannot be used. So going option wise among a,b,c .. u see max utility is achieved by c. But, can u find out any other combination of (x,y) at which utility is maximized even more than option c. ??
Also how to go about simultaneous move game??

vandita

On 26 Apr 2014 11:11, "subhayu [via Discussion forum]" <[hidden email]> wrote:

Vandita: Yes ur right if there is a square root then LPP cannot be used...did u check the question..??trying the second part..!!

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[Granpa Simpson]

yes ur right..but are u sure that its a square root...?

[vandita24x7]

Damn sure. Shelly made a typo. Its sqrt only

vandita

On 26 Apr 2014 11:23, "subhayu [via Discussion forum]" <[hidden email]> wrote:

yes ur right..but are u sure that its a square root...?

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[Granpa Simpson]

@Vandita: well then (0,25) or (25,0) or (12.5,12.5) will be the solution..!!!!

[shelly gupta]

hi subhayu...yes i made a typing mistake. its square root only. sorry .

[vandita24x7]

Yeah i got these. So shud i go with option c or d(None of above) is the confusion!!!!  What do u think??
Any clue abt q.3 ??

vandita

On 26 Apr 2014 11:51, "subhayu [via Discussion forum]" <[hidden email]> wrote:

@Vandita: well then (0,25) or (25,0) or (12.5,12.5) will be the solution..!!!!

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[Granpa Simpson]

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[vandita24x7]

I guess it shud be.. we can hv more than 1 nash equilibria isnt it?? Can u tell the exact defn of nash eqbm..

vandita

On 26 Apr 2014 11:56, "subhayu [via Discussion forum]" <[hidden email]> wrote:

In that case (0,25) will be the Nash equilibrium too..is it so..??

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[Granpa Simpson]

The point from which no player has incentive to move..even I think that there are multiple nash equilibria..!!!