# Competitive equilibirium of max utility functions

Apologies in advance if my terms aren't exact, I'm learning "Mathmatical Economoics" in the hebrew language and some of the terms don't translate well.

I was given the following question: Two sellers are a part of an exchange market with two goods, each has the utility function $$u(x,y)=\max(x,y)$$. Assume the first seller starts with two units of product $$x$$ and one unit of $$y$$.

Find $$(a,b)$$ s.t seller two starts with $$a$$ units of $$x$$ and $$b$$ units of $$y$$, for which there is a competetive equilebrium.

Any guidence will be of great help! Thanks in advance

• What are you confused about specifically? – Art Feb 25 '20 at 10:21

I might be wrong so feel free to point out any mistakes.

Both of them have the same utility function $$u(x,y)=max(x,y)$$ This means when $$Px > Py$$ , they'll choose to consume $$Y$$ instead of $$X$$ since they get higher satisfaction by selling all units of x and purchasing y.

Take the case of First seller, If he sells all X and purchases Y, His total demand for $$Y$$ will be $$2(Px/Py) + 1$$ and since Px/Py >1, total demand will be higher than 3. If he had done the opposite, his total demand will be less than 3. Since the utility function is $$max(x,y)$$ He will choose the former strategy

Hence, the demand for $$X$$ will be zero but the supply is $$a+2$$. Also, there is excess demand of $$Y$$. Hence, this is not a competitive equilibrium.

Similarly, then $$Px , there will be excess demand for $$X$$ so that isnt a competitive equilibrium either.

The only case we have left is $$Px=Py$$ where the seller will be indifferent in consuming either good. Here, each seller consumes all of 1 good each. There will be 2 cases.

Case 1. If seller 1 consumes all $$X$$, His demand is $$2 + (1)Py/Px = 3$$ since Px=Py and the supply of X is $$a+2$$. When we equate this, we get $$a=1$$ Notice that this is the only condition that needs to hold here, B can take any non negative values in this case.

Case 2 is where seller 1 consumes all $$Y$$ and, His demand is $$1+ (2)Px/Py = 3$$ and the supply of $$Y$$ is $$b+1$$. Since they have to be equal, we get $$b=2$$ And again there's no restriction on $$a$$ and it can take any non negative values.

• Thank you! I understand it now, finally :) – Gil Bar Koltun Feb 25 '20 at 10:55