General Equilibrium - June 17

[Amit Goyal]

June17__GE.png

[komal]

walrasian equilibrium for n=4 is 2 <p*<3 ( px/py) ,
i had earlier made mistake, consumption bundle would be , i guess, ((1- 1/p) , 2) for first 2 individual and ( (1+ 1/p ), 0 )

[anon_econ]

Komal, if these r the consumption bundles then to make total demand=total endowment, v will need p*=1. but at p*=1 these bundles wud nt be demanded in the first place.
i'm getting no walrasian equilibrium! plz correct me if i m wrong..and can't do the first one!

[komal]

u were right those bundles were incorrect

[rituvv]

yes vasudha i agree...we cant choose a price ratio so that all 4 can trade...apart from 1st person all others attach a higher index to x nd 1 is indifferent btwn them....for example if ratio is btwn 2nd 3 then individusl  3 rd nd 4 th person spend entire income on x nd dd > supply
1,2 on y nd dd> supply
so cant have a comp eqm in this case...

[Amit Goyal]

There is a Walrasian/Competitive equilibrium in this economy:
When price of X is Rs. 2/- and price of Y is Re. 1/-.
Individual 1 will consume only Y: (0, 3)
Individual 2 will consume his endowment: (1, 1)
Individual 3 will consume only X: (1.5, 0)
Individual 4 will also consume only X: (1.5, 0)

[komal]

@ amit sir, this is only one of the equilibrium, right?

[Amit Goyal]

This is the only equilibrium.

[anon_econ]

Thanks. I have understood my mistake :)

[anon_econ]

But what about pareto efficiency?

[komal]

i have got really confused, if there is a question that there r 2 individauals with u = 3x1 + y1 and 4x2 +y2,respectively and endowmwnts (1,1) each , then what will the the walrasian equilibrium?

[Amit Goyal]

Komal, In the economy you mentioned Competitive equilibrium prices are:
Price of X is 3 and price of Y is 1.
Individual with utility function u(x, y) = 3x + y consumes the following bundle: (2/3, 2)
Individual with utility function u(x, y) = 4x + y consumes only X: (4/3, 0)

[Amit Goyal]

Vasudha, for set of Pareto efficient allocations, I suggest you do it for the cases N = 2 and N = 3 first. You will observe a pattern. And then you can try and prove it for general N.

[ritu]

sir for N=2 i got PE set as the vertical axis of agent 1 and horizontal axis of agent 2.....how to get it for 3 individuals....?????

[AJ]

ummmm, i gave my best shot to it....and it's weird answer :/

I am getting pareto efficient allocation something like...

for Nth person, (x,y)= (N,y) or (x,0) ... x, y belongs to [0,N]

Now,
~ When this person is having.. (N,y) .. the remaining can have any distribution of good Y.. and allocation will be pareto optimal.

~ when this person is having .. (x,0) ... the (N-1)th person should have .. either (N-x, y) or (x',0)..
                                                      ~ when he has.. (N-x, y) ... the remaining can have any distribution  of good Y
                                                      ~ when he has.. (x',0) ... the (N-2)th person should have... either (N-x-x') or (x'',0)

...............
and now we continue same way till 1st person...

I am basically trying to say that last person should either have "all X" or "no Y" .... if he has all X... all allocations are optimal.. if he has no Y(and some X).. we move to previous person..

Now he must have "all remaining X" .. or "no Y" .... if 1st case: all allocations b/w remaining people are optimal.... if 2nd case: we move to previous person

Now he must have "all remaining X" .. or "no Y" ... and continues.......

[komal]

 p=4 can also be equilibrium
where  first individual with utility 3x + y consumes (3/4, 2)
 and second with utility = 4x +y consumes (5/4,0) ,
 am i right ?
 i m not able to understand why is there only one walrasian equilibrium ,  even this appears to me as walrasian equilibrium. where m i wrong ?

[ritu]

komal at p=4.....individuals 1,2,3 all will choose to consume only y as for all of them mrs<p1/p2....dd>supply....hence not an equlm

[ritu]

individual 1 has x+y
2 has x+2y

[komal]

@ritu, actually i m refering to a question asked in between, plz help me out if u know..

[anon_econ]

y do u say he wud consume (3/4,2)?