DSE-2010 Ques: 31-34

[Arun]

I  tried solving the problem but still have doubts.I read Equilibrium part from both Pindyk and Varian but still confused.Given below are my solutions  and any pointers would be appreciated.


 31) (D)  1 gets (1,0) and 2 gets (1,1)
As it is a pareto efficient solution


32) (A) Px/Py = 1

 (1,0)  and (1,1) would be the pareto efficient solution and the exchange rate of X:Y is 1:1
Or
alternately Agent 1 can offer maximum 1 y for x and Agent would won't trade 1 x for less than 1 y so the agreeable solution is 1:1

33)The distribution is already pareto efficient so there is no scope of trade.How do we decide the price ratio??

34)Initial distribution between agent 1 and 2 is (0,1) and (2,0).As Agent 2 treats x and y as perfect complements so he would be indifferent between (2,0) and (0,0) therefore should not (2,1) and (0,0) be a competitive equilibrium allocation?



Any pointer to some text which could help clear my doubts would be appreciated.

[Ayushya Kaul]

Bump.

Could anyone answer these doubts? Have the same ones!

[laracroft]

31- D


32- A


33- C


34- C

[maahi]

how?? for 33 ,34

[Sumit]

Hey laracroft,
would you mind to share your approach????
 

[madhur1987]

in the que 33 endowment bundle is pareto efficient(even thogh agent1 won't trade but agent2 can still trade he is indiff) so to make sure that there is no trade price ratio should be like something that even agent 2 do not want to trade this will be only possible when Px<=Py. so no trade is possible

[madhur1987]

in que 34 (2,1) i also thin is competetive equillibrium but option d is saying all the allocations not only (2,1) thats why option d might not be correct... please someone corrects me if i am wrong

[Ayushya Kaul]

Yeah, I can't reason out question 34.

[Prerna Rakheja]

Hi Madhur,

For; Agent 1 gets (2,1) and Agent 2 gets (0,0) to be a competitive equilibrium it must be that px/py = 0 ==> px = 0

Since agent 1 lexicographically prefers y to x, thus after having attained all the y that is possible given that x is a free good he would demand infinite amount of x, while total endowment is only 2 units. Hence, market wont clear thus this cannot be a competitive equilibrium.

[duck]

Hi.. :)

31) Option(d)
32) Option(a)
Reasoning:
For any price ratio: Agent 1 demand would be: x1=py/px ; y1=0
And for agent 2: If px>py then, x2=0 and y2= 2px/py
                       If px<py then, x2=2 and y2=0
                       If px=py then, {(x2,y2)| x2+y2=2}

Clearly, Market is not clearing for px<py and px>py.
But when px=py then, At (x1=1, y1=0) and (x2=1 , y2=1) clears the market.
So, Competitve equilibirum Allocation is: Agent 1 getting (1,0) and Agent 2 getting (1,1).

33) Now, Agent1 lexicographically prefers y over x.
So, for all price ratio his demand would be: x1=0 and y1=1
Demand for Agent 2 remains the same as above.
Now, when price ratio ≤ 1 then, market is clearing.
Hence, option(c)

34) Agent 1 demand : x1=0 and y1=1 for all strictly positive price ratio.
And as agent2 treats both goods as perfect complements. He would consume the same amount of x and y or all price ratio i.e x2=y2= 2px/px+py
Now, for market clearing we need x2=2 and y2 to be 1 implying x2≠y2.
For (py=0 and px>0) or (px=0 and py>0), agent1 would demand infinite amounts. So, again market will not clear.
Therefore, no competitve equilibirum exists.

[Sumit]

@Duck:Thanks a lot ...You explained it in such a simple manner.

But I have a small doubt in Q34...I able to get my answer correct simply looking the demand and supply for good y. (that is agent-2 to will now demand 1 unit of Y but agent-1 will not supply it) so,there will be excess demand of Y...so, market is not clearing.Therefore, no competitve equilibirum exists...My question is the approach I used here is correct or not????...Actually I asking you this question bcoz I able to get all the answer correct simply assuming Py=1...and then looking which price of X..will clear the market in all the cases...

One more thing...In second last line of your answer---you mentioned For (py=0 and px>0) or (px=0 and py>0), agent1 would demand infinite amounts...
I agree for (py=0 and px>0)...agent1 demand will be infinite for y (coz he is actually getting good y for free) and zero for x...
but for (px=0 and py>0) would not be agent1 demand for x would be infinite and for y=1?????

and Thanks once again...Its mean a lot to me...



 

[duck]

Hi Sumit.. :)

For Q34) Agent2's demand is y=2px/px+py and not just 1.
You must have assumed px=1 along with py=1 and then, you got Agent2's demand as one unit of y.
So, be careful.  
Also,there is no problem in assuming one price to be a numeraire price and checking for other price.. :)

And for (px=0 and py>0) ,
Agent1: Max "y" he can afford is 1 at these prices and demand infinite "x" as its freely available.

[Sumit]

Got it..thanks..:-D