DSE Q.NO-34 2010

[sweata]

34- Now suppose the agent 1 lexicographically prefers y to x and agent 2 treats x and y as perfect complements.the set of competitive equilibrium allocations.

ans provided is empty but if we draw the Edgeworth box the initial endowment of individual 1 WITH LEXICOGRAPHIC PREFERENCE IS (1,0) and that of individual two is (2,0) now if we move from (2,0) to (2,1) individual one is better off because with any two bundles having the same amount of y he would prefer the bundle with more x.but individual two would be indifferent.this point also occurs at the kink of the l shaped preference of individual two.why is this bundle not Pareto optimal.PARTICULARLY I WANT TO KNOW WHY IS IT EMPTY?I cant understand the logic.

[sweata]

i mean initial endowment of 1 is (0,1) and if we move to (2,1) then individual 2 has (0,0) ..why is this not a competitive equilibrium

[Amit Goyal]

Reading what you wrote i don't think you know the meaning of competitive equilibrium. You are confusing the notion of efficiency with equilibrium. Please read the book to know what an equilibrium means and then solve this problem.

[Arun]

Even i had some confusion over this question.My line of reasoning is

1)Initial allocation is not competitive equilibrium as it is not pareto efficient because allocation (2,1) (0,0) is possible which increases Agent 1 utility and keeps Agent 2 utility constant.


2)A competitive equilibrium should be pareto efficient.Allocation (2,1) (0,0) could have been a competitive equilibrium but its not reachable.Agent 2 wont trade 'x' at a price zero.Therefore given the initial distribution and preferences no trade is possible and initial distribution is not pareto efficient so the result is a empty set.


I am not very confident so it would be preferred that someone could verify the above reasoning.

[Amit Goyal]

Well If you are using First Welfare Theorem for solving the problem then you need to check for efficiency of the options. If you cannot rule out certain option by efficiency argument then you need to use the definition of equilibrium to check. If this is what you are doing you are right.

[poonam]

 Following id the approach i used... kindly let me know if its correct.
general equilibrium is a pareto efficient allocation , for a "given" price level.
So price ratio will be the slope of the line passing through the initial endowment and the pareto efficient allocation.

by the same logic... the movement from initial endowment to (2,1)  (0,0) will not take place because that is possible only if the price of good 1 is equal to zero.

and the initial allocation is not pareto efficient. so the answer becomes null set.

would the above approach be applicable everytime ??

[Amit Goyal]

I think your approach is based on your knowledge of the answer. But its not that you will know the answer in exam. If you try and generalize the approach you might end up in trouble. I suggest find CE by first finding demand functions and then replace incomes by the value of the endowments and then you may use market clearing condition to solve for prices. This is the simplest and the best way.

[poonam]

thanks...
but what would be the demand function for lexicographic preferences ?? moew so .. what would be teh utility function for the same ?? so in this case how will we use the standard competitive equilibrium approach ??

[Amit Goyal]

Poonam, Given that the price of x is positive (p_x > 0) and price of y is positive (p_y > 0), if you lexicographically prefers x to y then the demand function for x = M/p_x, y =0 (Think why?)

[poonam]

ohk.... thank you sir, I got it . :-) 

On Fri, Jun 24, 2011 at 5:47 AM, Amit Goyal [via Discussion forum] <[hidden email]> wrote:

Poonam, Givne that the price of x is positive and price of y is positive, if you lexicographically prefers x to y then the demand function for x = M/p_x, y =0 (Think why?)

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[vikram]

In the same question set, I have a doubt in the previous question where agent 1 lexicographically prefers y to x and agent 2 treats x and y as perfect substitutes, and we need to find the set of competitive equilibrium prices p_x and p_y (both > 0).

Here, the competitive equilibrium allocation should remain the same as initial endowment. Walras law doesn't seem to work. Please help as to where I am going wrong.