<p>Given the exchange economy,</p>
<ul>
<li>Three Goods: X (Apple), Y (Banana), M (Money)</li>
<li>Three Agents: Seller (S), Buyer 1 (A), Buyer 2 (B)</li>
<li>Endowment of the seller: <span class="math-container">$W_S=(1,1,0)$</span>; Endowment of the buyers: <span class="math-container">$W_A=W_B=(0,0,10)$</span>. Seller has apple and banana, buyers have the money.</li>
<li>Utility functions: For <span class="math-container">$i\in\{S,A,B\}$</span>, define <span class="math-container">$u_i:\{0,1\}\times\{0,1\}\times\mathbb{R}_+\rightarrow\mathbb{R}$</span> as follows:</li>
</ul>
<p><span class="math-container">$u_S(x_S,y_S, m_S)=m_S$</span>,</p>
<p><span class="math-container">$u_A(x_A,y_A, m_A)=3\min(x_A,y_A)+m_A$</span>,</p>
<p><span class="math-container">$u_B(x_B,y_B, m_B)=2\max(x_B,y_B)+m_B$</span></p>
<p>We can quickly rule out the possibility of having <span class="math-container">$p_M=0$</span> because that would lead to unbounded demand for <span class="math-container">$M$</span>.</p>
<p>So, we let <span class="math-container">$p_M=1$</span> be the numeraire. We'll now consider the following possibilities:</p>
<ul>
<li><p><span class="math-container">$p_X=0$</span> and <span class="math-container">$p_Y=0$</span>. Seller is indifferent between selling and buying apples and bananas, but given that both are free, A will demand 1 unit of apple and 1 unit of banana, and B will also demand either 1 unit of apple or 1 unit of banana. So, at least one of the two will be in excess demand.</p>
</li>
<li><p><span class="math-container">$p_X=0$</span> and <span class="math-container">$p_Y&gt;0$</span>. B will demand 1 unit of X. A will definitely demand X and Y in such a way that quantity demanded of X is at least as much as quantity demanded of Y. So the aggregate demand for X will be at least one unit plus the aggregate demand for Y. However, at these prices, the supply of X will be at most as much as the supply of Y. Therefore, markets will not clear.</p>
</li>
<li><p><span class="math-container">$p_X&gt;0$</span> and <span class="math-container">$p_Y=0$</span>. This also ruled out by symmetric argument as in the case before.</p>
</li>
<li><p><span class="math-container">$p_X&gt;0$</span> and <span class="math-container">$p_Y&gt;0$</span>. Since <span class="math-container">$p_X&gt;0$</span> and <span class="math-container">$p_Y&gt;0$</span>, observing the seller's utility function we know that seller would want to sell both the goods irrespective of the prices. So the supply of X is 1 and supply of Y is 1.</p>
</li>
</ul>
<p>Case 1: <span class="math-container">$p_X+p_Y&gt;3$</span>, In this case, A's demand is <span class="math-container">$x^d_A=y^d_A=0$</span> and B will demand either just <span class="math-container">$1$</span> unit of X, or just <span class="math-container">$1$</span> unit of Y, or <span class="math-container">$0$</span> unit of both X and Y depending on the prices. No matter what happens both the markets never clear at the same time.</p>
<p>Case 2: <span class="math-container">$p_X+p_Y&lt;3$</span>, In this case, A's demand is <span class="math-container">$x^d_A=y^d_A=1$</span> and B will demand either just <span class="math-container">$1$</span> unit of X, or just <span class="math-container">$1$</span> unit of Y depending on whichever is cheaper. This is because <span class="math-container">$p_X+p_Y\leq 3$</span> implies <span class="math-container">$\min(p_X,p_Y)&lt;2$</span>. Again no matter what happens there will be excess demand for the cheaper of the two goods, or for one of them if they are equally priced.</p>
<p>Case 3: <span class="math-container">$p_X+p_Y=3$</span>, In this case A can either demand 0 unit of both, or 1 unit of both. If A demand 0 unit of both then Case 1 reasoning applies. If A demand 1 unit of both the goods then Case 2 reasoning applies. Either ways, competitive equilibrium does not exist.</p>

CC BY-SA 4.0
2023-12-06T14:41:26.590Z
This is correct
High GPA
CC BY-SA 4.0
2023-12-06T11:26:51.887Z
2023-12-06T13:28:37.740Z
2023-12-06T13:28:37.740Z
Amit
<p>There are 3 agents, one seller and two buyers. There are two indivisible goods, apple(a) and banana(b), and one divisible good, money(m). The seller's endowment is <span class="math-container">$W_s=(1,1,0)$</span>, and buyers's are <span class="math-container">$W_{b1}=W_{b2}=(0,0,10)$</span>. The utility function of the seller is <span class="math-container">$U_s(a,b,m)=m$</span>. For buyer 1, <span class="math-container">$u_{b1}(1,1,m)=3+m$</span>, and <span class="math-container">$u_{b1}(a,b,m)=m$</span> if <span class="math-container">$(a,b)\neq(1,1)$</span>. For buyer 2, <span class="math-container">$u_{b2}(a,b,m)=2+m$</span> if <span class="math-container">$(a,b)\neq(0,0)$</span> and <span class="math-container">$u_{b2}(0,0,m)=m$</span>.</p>
<p>I do not know how to show this economy does not have a WE.</p>

CC BY-SA 4.0
2023-12-05T09:28:50.367Z
Have you calculated how much each buyer buys at each price?
Michael Greinecker
CC BY-SA 4.0
2023-12-07T06:38:27.257Z
Yes, it is the same as what Amit answered.
user45481
<walrasian>
How to show there is no a Walrasian equilibrium?
user45481
# How to show there is no a Walrasian equilibrium?

(Asked by: username_0 on 2023-12-05)

There are 3 agents, one seller and two buyers. There are two indivisible goods, apple(a) and banana(b), and one divisible good, money(m). The seller's endowment is $W\_s=(1,1,0)$, and buyers's are $W\_{b1}=W\_{b2}=(0,0,10)$. The utility function of the seller is $U\_s(a,b,m)=m$. For buyer 1, $u\_{b1}(1,1,m)=3+m$, and $u\_{b1}(a,b,m)=m$ if $(a,b)\neq(1,1)$. For buyer 2, $u\_{b2}(a,b,m)=2+m$ if $(a,b)\neq(0,0)$ and $u\_{b2}(0,0,m)=m$.

I do not know how to show this economy does not have a WE.

**Tags:** walrasian
> **Comments:**
> **username_1**: Have you calculated how much each buyer buys at each price?  
> **username_0**: Yes, it is the same as what Amit answered.  

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## 1 answers

### Answer by username_2 on 2023-12-06. Score: 3 | Accepted answer

Given the exchange economy,

* Three Goods: X (Apple), Y (Banana), M (Money)
* Three Agents: Seller (S), Buyer 1 (A), Buyer 2 (B)
* Endowment of the seller: $W\_S=(1,1,0)$; Endowment of the buyers: $W\_A=W\_B=(0,0,10)$. Seller has apple and banana, buyers have the money.
* Utility functions: For $i\in\{S,A,B\}$, define $u\_i:\{0,1\}\times\{0,1\}\times\mathbb{R}\_+\rightarrow\mathbb{R}$ as follows:

$u\_S(x\_S,y\_S, m\_S)=m\_S$,

$u\_A(x\_A,y\_A, m\_A)=3\min(x\_A,y\_A)+m\_A$,

$u\_B(x\_B,y\_B, m\_B)=2\max(x\_B,y\_B)+m\_B$

We can quickly rule out the possibility of having $p\_M=0$ because that would lead to unbounded demand for $M$.

So, we let $p\_M=1$ be the numeraire. We'll now consider the following possibilities:

* $p\_X=0$ and $p\_Y=0$. Seller is indifferent between selling and buying apples and bananas, but given that both are free, A will demand 1 unit of apple and 1 unit of banana, and B will also demand either 1 unit of apple or 1 unit of banana. So, at least one of the two will be in excess demand.
* $p\_X=0$ and $p\_Y>0$. B will demand 1 unit of X. A will definitely demand X and Y in such a way that quantity demanded of X is at least as much as quantity demanded of Y. So the aggregate demand for X will be at least one unit plus the aggregate demand for Y. However, at these prices, the supply of X will be at most as much as the supply of Y. Therefore, markets will not clear.
* $p\_X>0$ and $p\_Y=0$. This also ruled out by symmetric argument as in the case before.
* $p\_X>0$ and $p\_Y>0$. Since $p\_X>0$ and $p\_Y>0$, observing the seller's utility function we know that seller would want to sell both the goods irrespective of the prices. So the supply of X is 1 and supply of Y is 1.

Case 1: $p\_X+p\_Y>3$, In this case, A's demand is $x^d\_A=y^d\_A=0$ and B will demand either just $1$ unit of X, or just $1$ unit of Y, or $0$ unit of both X and Y depending on the prices. No matter what happens both the markets never clear at the same time.

Case 2: $p\_X+p\_Y<3$, In this case, A's demand is $x^d\_A=y^d\_A=1$ and B will demand either just $1$ unit of X, or just $1$ unit of Y depending on whichever is cheaper. This is because $p\_X+p\_Y\leq 3$ implies $\min(p\_X,p\_Y)<2$. Again no matter what happens there will be excess demand for the cheaper of the two goods, or for one of them if they are equally priced.

Case 3: $p\_X+p\_Y=3$, In this case A can either demand 0 unit of both, or 1 unit of both. If A demand 0 unit of both then Case 1 reasoning applies. If A demand 1 unit of both the goods then Case 2 reasoning applies. Either ways, competitive equilibrium does not exist.

> **Comments:**
> **username_3**: This is correct  

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