<p>I might be wrong so feel free to point out any mistakes.</p>

<p>Both of them have the same utility function <span class="math-container">$u(x,y)=max(x,y)$</span>
This means when <span class="math-container">$Px &gt; Py$</span> , they'll choose to consume <span class="math-container">$Y$</span> instead of <span class="math-container">$X$</span> since they get higher satisfaction by selling all units of x and purchasing y.</p>

<p>Take the case of First seller, If he sells all X and purchases Y, His total demand for <span class="math-container">$Y$</span> will be <span class="math-container">$2(Px/Py) + 1$</span> 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 <span class="math-container">$max(x,y)$</span> He will choose the former strategy</p>

<p>Hence, the demand for <span class="math-container">$X$</span> will be zero but the supply is <span class="math-container">$a+2$</span>. Also, there is excess demand of <span class="math-container">$Y$</span>. Hence, this is not a competitive equilibrium.</p>

<p>Similarly, then <span class="math-container">$Px &lt;Py$</span>, there will be excess demand for <span class="math-container">$X$</span> so that isnt a competitive equilibrium either.</p>

<p>The only case we have left is <span class="math-container">$Px=Py$</span> where the seller will be indifferent in consuming  either good. Here, each seller consumes all of 1 good each. There will be 2 cases.</p>

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

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

CC BY-SA 4.0
2020-02-25T10:55:48.427Z
Thank you! I understand it now, finally  :)
Gil Bar Koltun
CC BY-SA 4.0
2020-02-25T10:42:54.153Z
2020-02-25T10:42:54.153Z
Abhi Minhas
<p>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.</p>

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

<p>Find <span class="math-container">$(a,b)$</span> s.t seller two starts with <span class="math-container">$a$</span> units of <span class="math-container">$x$</span> and <span class="math-container">$b$</span> units of <span class="math-container">$y$</span>, for which there is a competetive equilebrium. </p>

<p>Any guidence will be of great help! Thanks in advance</p>

CC BY-SA 4.0
2020-02-25T10:21:23.277Z
What are you confused about specifically?
Art
<competitive-equilibrium>
Competitive equilibirium of max utility functions
Gil Bar Koltun
# Competitive equilibirium of max utility functions

(Asked by: username_0 on 2020-02-24)

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

**Tags:** competitive-equilibrium

> **Comments:**
> **username_1**: What are you confused about specifically?  

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## 1 answers

### Answer by username_2 on 2020-02-25. Score: 1 | Accepted answer

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 $Px <Py$, 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.

> **Comments:**
> **username_0**: Thank you! I understand it now, finally  :)  

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