Re: DSE 2006

[poonam]

Please help me with the following question:

Consider a competitive exchange economy with 2 agents (1 and 2) and 2 goods (X and Y). Agent 1's endowment is (0,5) and Agent 2's endowment is (0,10). Agent i's objective is to chose (xi, yi)  to maximize his utility min{xi, yi}.

Q.23 Assuming the sum of prices is 1 , then the competitive equilibrium prices are ??

[poonam]

This is how i tried to solve it .. but i have some conceptual issues... Please guide me forth.

Using the competitive equilibrium approach and taking no numeraire, because there is an additional relation in the question.
we get M1=5*p_2
          M2=10*p_1

Dx_1 = Dy_1= M1/(p_1 + P_2)= M1

similarly, Dx_2 = Dy_2 = M2

so we get Dx_1 + Dx_2 = 10  -------(i)             and Dy_1 + Dy_2 = 5   ---------(ii)
the above two equations simultaneously have no solution.
solving each with p_1 + p_2 =1 ------(iii)
we get two set of prices : (1,0) { from (i) and (iii) }
                                     (0,1) { from (ii) and (iii) }

Now what ???
how do we choose between the two ??

And this is when we try to solve it like this, if we use the numeraire approach, then the plain choice of numeraire gives us two different answers. What is the criteria to choose ?
Please let me know ...

[Amit Goyal]

To solve this problem it is not sufficient to just write the demand for positive prices. You need to write it for zero prices as well.
If u(x, y) = min{x, y} then demand is given by
x(p_x, p_y, M) = M/(p_x+p_y) for p_x>0, p_y>0
=[M/p_y, infinity) for p_x = 0, p_y>0
The idea is when p_x= 0 and p_y>0 you will consume at least as many number of units of x as you can buy y. Use this and then solve it.

[poonam]

Two questions please:
a) how does this change the situation??
b) if we consider p_x=0 , p_y>0,  then we also keep open the possiblility of p_x>0 and p_y=0, right ???

[Amit Goyal]

ok let me write the complete demand function
x(p_x, p_y, M) = M/(p_x+p_y) for p_x>0, p_y>0
=[M/p_y, infinity) for p_x = 0, p_y>0
=M/p_x for p_x>0, p_y=0

[poonam]

Thank you sir. That answers the second doubt however the fist doubt still remains unsolved.
 I still cant get it how we decide to reject (1,0) and accept (0,1).

I would be thankful if you could elaborate on that , please..

[Amit Goyal]

Consider a competitive exchange economy with 2 agents (1 and 2) and 2 goods (X and Y). Agent 1's endowment is (0,5) and Agent 2's endowment is (10,0). Agent i's objective is to chose (xi, yi)  to maximize his utility min{xi, yi}.

Using the demand functions, we get the following:
Lets compute demands at prices (1,0) first
Individual 1 demands 0 units of good x and at least 0 units of good y.
Individual 2 demands 10 units of good x and at least 10 units of good y.

Market clearing condition fails because at least 10 units of good y are demanded and we only have 5.

Lets compute demands at prices (0,1) now
Individual 1 demands at least 5 units of good x and 5 units of good y.
Individual 2 demands at least 0 units of good x and 0 units of good y.

So market can clear when 1 demands (x, 5) and 2 demands (10-x, 0) for any 5<=x<=10.

[poonam]

thank you so much sir...