Amit sir plz help...

[Sumit]

situation:An exchange economy is populated by three consumers: 1, 2, and 3. There are two goods—good x and good y. The consumers’ preferences are characterized (respectively) by
the three utility functions: u(1) = min{x, y}, u(2) = 2log(x) + y, u(3) = x + 2y
Total endowment of good x and y in the economy is 10 units each.


q1)Which of the following allocations is pareto efficient?
a)((1, 1), (1, 0), (8, 9))
b)((4, 4), (1, 2), (5, 4))
c) ((2, 2), (8, 0), (0, 8))
d)((1, 1), (0, 1), (9, 8))

q2)
Suppose the endowment is the pareto efficient allocation of the problem above. Then competitive equilibrium prices are
 a) (1, 1)
 b)(1, 3)
 c)(1, 4)
  d)None of the above


Sir plz tell how to go about Q2...


N the second part of trade question in the following  link....

http://discussion-forum.2150183.n2.nabble.com/TRADE-tp7580461.html

[Devika]

Is it part a for q1?

[Sumit]

Right Devika...

[Devika]

Sumit, what was your approach towards the question? I did the usual transfer method and then compared the resulting utilities. But it took a toll over me, as there are 4 parts and three utilities to be compared. Do you know of any method that's not this cumbersome?

[Sumit]

Hey Devika,
here is how I did...http://discussion-forum.2150183.n2.nabble.com/doubt-P-O-and-Gen-eq-tp7581810.html
...I think the only way to solve this question is the same as you mentioned....

[Devika]

Great. Thank you! :)

[snigdha]

hey can u guys explain hw u got a) as the ans..
the utilities come out to be,
a) u1 = 1    u2 =0     u3=26
d) u1 = 1    u2 = 1    u3 = 25

why isnt d) pareto efficient?

[duck]

Hi Snigdha.. :)

(d) is not pareto optimal.
at (d), Utility of agent 2 is not "1".
log(x) tends to "minus infinity" as x tends to zero.
So, his utility will be negative infinity or not defined.
Hence, you can reallocate in the following way:
Agent1:(1,1)
Agent2: (1,0)
Agent3: (8,9)

[duck]

Hi .. :)

For competitive equilibirum, find the demand for all three individuals and then check at what price ratio market clears.

Answer: None of the above.
Competitive equilibirum Price ratio = 1/2.
At (1,1): x1=y1=1; x2=1,y2=0 and x3=0,y3= 15
At (1,3): x1=y1=1; x2=1,y2=0 and x3=35 , y3=0
At (1,4): x1=y1=1; x2=1,y2=0 and x3=44 , y3=0.

Market is not clearing at any above price ratio.

[Sumit]

thanks a lot duck...you saved me.........

Also At(1,1) would not be x3=0, y3=17...instead of 15???

[duck]

Yup.. 17.. :)

[madhur1987]

in the solution given by duck why is X2=1 and Y2=0 at all the prices? just bcoz his m is 1 and thats the max he can afford?

[Sumit]

Right madhur...In all the above case bcoz Price of x=1...N due to which his M=1...So, Max he can afford is the one unit of x...

[madhur1987]

thanx sumit :)

[Ayushya Kaul]

Hey duck,
Could you please show me how you obtained the budget lines for the consumers in the 2nd part of the question?
I'm confused :s

[duck]

Hi.. :)
Endownment is {(1,1); (1,0); (8,9)}
So, Income of Agent1: px+py
Income of Agent2: px
Income of Agent3: 8px+9py.

Now, its easy to figure out the budget lines:
Agent1: x.px+y.py=px+py
Likewise for other agents.