DSE 2006 - Q.21, 22, 23

[Swati Kanodia]

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. 21 The allocation (x1, y1) = (3,3) and (x2, y2) = (7,2) is
I was able to solve this part considering that this allocation lies in the pareto band but given agent 2's endowment he would not like to reach here. thus, it is pareto effecint but not competitive equilibrium.

Q.22 The allocation with (x1, y1) = (10,5) and (x2, y2) = (0,0) is

The answer says that it is a competitive equilibrium and is pareto effecient.

I could understand the reason for pareto effeciency but not competitive equilibrium.

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

a) (1,0)
b) (0,1)
c) (.5, .5)
d) (1/3, 2,3)

Also there is a doubt that if i consider that the budget line must pass through the endowment and thus it should be horizontal i am able to solve Q.22 but then by this logic can not the BL be vertical also??????

Please clear my doubt...

[Amit Goyal]

Forget about competitive equilibrium for the time being.
First answer this problem: A consumer with utility function u(x, y) = min{x, y} faces price of x = 1 and price of y = 0. Suppose he is a utility maximizing consumer and has income 100 to spend on the two goods. Find the set of all solutions to the utility maximization problem (subject to budget constraint and non-negativity constraints).

[poonam]

With respect to Q23:
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
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 ...

[poonam]

p_1 + p_2 = 1 ----(iii)