<p>Given an exchange economy:</p>
<ul>
<li>Two consumers - <span class="math-container">$u_1(x_1,y_1)=Ax_1+y_1$</span>, where <span class="math-container">$A&gt;0$</span> is given, <span class="math-container">$u_2(x_2,y_2)=x_2y_2$</span>.</li>
<li>Endowments - <span class="math-container">$\omega_1=(e,0)$</span>, <span class="math-container">$\omega_2=(0,e)$</span>, where <span class="math-container">$e&gt;0$</span> is given.</li>
</ul>
<p>Set of feasible allocations is <span class="math-container">$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=y_1+y_2=e\}$</span></p>
<p>Set of Pareto efficient allocations is</p>
<p><span class="math-container">$\mathcal{PE}=\begin{cases} \{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|y_2=\min(Ax_2,e)\} &amp; \text{if } A \geq 1 \\ \{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|(y_2=Ax_2) \ \vee \ (y_2\geq Ae \ \wedge \ x_2=e) \} &amp; \text{if } A &lt; 1\end{cases}$</span></p>
<p>Competitive equilibrium allocation and the supporting price is</p>
<p><span class="math-container">$(((x_1^c,y_1^c),(x_2^c,y_2^c)),(p_X^c,p_Y^c))=\begin{cases} \left(\left(\left(\frac{(2A-1)e}{2A},\frac{e}{2}\right),\left(\frac{e}{2A},\frac{e}{2}\right)\right),(A,1) \right) &amp; \text{if } A\geq\frac{1}{2} \\ \left(\left(\left(0,\frac{e}{2}\right),\left(e,\frac{e}{2}\right)\right),(\frac{1}{2},1) \right) &amp; \text{if } A&lt;\frac{1}{2} \end{cases}$</span></p>

CC BY-SA 4.0
2024-10-04T17:44:09.213Z
2024-10-04T17:44:09.213Z
Amit
<p>We have the following information,</p>
<p><span class="math-container">$u_1(x_{11},x_{12})=A\cdot x_{11}+x_{12}$</span></p>
<p><span class="math-container">$u_2(x_{11},x_{12})=x_{21}\cdot x_{22}$</span></p>
<p><span class="math-container">$e_1=(e,0)$</span></p>
<p><span class="math-container">$e_2=(0,e)$</span></p>
<p>Solving Agent 1's utility maximization problem,
<span class="math-container">$$\begin{eqnarray}
\max_{{(x_{11},x_{12}}) \ \in \ \mathbb{R}^2_+} &amp; A\cdot x_{11}+x_{12} \\
&amp; s.t. p_1x_{11}+p_2x_{12}\leq p_1e
\end{eqnarray}$$</span>
where <span class="math-container">$p_1e$</span> is the value of agent 1's endowment, we get the agent 1's demand correspondence as,
<span class="math-container">$$\begin{eqnarray}
(x_{11}^d,x_{12}^d)\in 
\begin{cases}
\{(0,\frac{p_1}{p_2}e)\} &amp; \text{if} \ A &lt; \frac{p_1}{p_2}\\
\{(e,0)\} &amp; \text{if} \ A &gt; \frac{p_1}{p_2}\\
\{(x_{11},x_{12})\in \mathbb{R}^2_+ \ | \ p_1x_{11}+p_2x_{12}=p_1e\}
&amp; \text{if} \ A = \frac{p_1}{p_2}\\
\end{cases}
\end{eqnarray}
$$</span>
Solving Agent 2's utility maximization problem,
<span class="math-container">$$\begin{eqnarray}
\max_{{(x_{21},x_{22}})\ \in \ \mathbb{R}^2_+} &amp;  x_{11} \cdot x_{12} \\
&amp; s.t. p_1x_{21}+p_2x_{22}\leq p_2e
\end{eqnarray}$$</span>
where <span class="math-container">$p_2e$</span> is the value of agent 2's endowment, we get the agent 2's demand function as,
<span class="math-container">$$(x_{21}^d,x_{22}^d)=\left(\frac{p_2e}{2p_1},\frac{e}{2} \right)$$</span></p>
<p>let's take price of good 2 as the numeraire and then our demand functions take the form,
<span class="math-container">$$\begin{eqnarray}
(x_{11}^d,x_{12}^d)\in 
\begin{cases}
\{(0,p_1e)\} &amp; \text{if} \ A &lt; p_1\\
\{(e,0)\} &amp; \text{if} \ A &gt; p_1\\
\{(x_{11},x_{12})\in \mathbb{R}^2_+ \ | \ p_1x_{11}+x_{12}=p_1e\}
&amp; \text{if} \ A = p_1\\
\end{cases}
\end{eqnarray}
$$</span>
and
<span class="math-container">$$(x_{21}^d,x_{22}^d)=\left(\frac{e}{2p_1},\frac{e}{2} \right)$$</span></p>
<p>Now we can solve for the competitive equilibrium of this economy, I will assume WLOG that <span class="math-container">$A&gt;0$</span>,
so competitive equilibrium price vector will be dependent on the different values that <span class="math-container">$A$</span> takes so I will consider three different cases for the same,</p>
<p>Case 1, <span class="math-container">$\ 0&lt;A&lt;1$</span>, Case 2, <span class="math-container">$\ A=1$</span>,and Case 3, <span class="math-container">$\ A&gt;1$</span></p>
<p>So we have the following equilibrium price vector,
<span class="math-container">$$
(p_1,p_2)=(A,1)
$$</span>
Here is the example of how the Edgeworth Box would look like for the case when <span class="math-container">$A=\frac{p_1}{p_2} = 1$</span>
<a href="https://i.sstatic.net/M6kzLjopm.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/M6kzLjopm.png" alt="enter image description here" /></a></p>
<p>Intuitively one can reason out that <span class="math-container">$(p_1,p_2)=(A,1)$</span> is the competitive equilibrium price vector since when <span class="math-container">$\frac{p_1}{p_2}=A$</span>, agent 2 will maximise his utility and given the budget constraint and the remaining quantities will be given to agent 1, and for any other price vector other that this there will be excess demand or excess supply of either good in the economy.</p>

CC BY-SA 4.0
2024-10-03T15:48:43.020Z
2024-10-04T18:23:53.803Z
2024-10-04T18:23:53.803Z
econhead
<p>Maybe the title doesn't reflect what I mean perfectly but basically, I wanna derive demand functions from those two utility functions:
<em>where  <span class="math-container">$x_{11}$</span> is the consumption of good 1 by agent 1 and  <span class="math-container">$x_{12}$</span> is the consumption of good 2 by agent 1 and so on.</em>
<span class="math-container">$$
u_1(x_{11},x_{12}) = A \cdot x_{11}+x_{12}
$$</span></p>
<p>and endowments of agent 1 are <span class="math-container">$e_1 = (e,0)$</span></p>
<p><span class="math-container">$$
u_2(x_{21},x_{22}) = x_{21}\cdot x_{22}
$$</span></p>
<p>and endowments of agent 2 are <span class="math-container">$e_2 = (0,e)$</span></p>
<p>When <span class="math-container">$A&gt;p_1/p_2$</span> or <span class="math-container">$A&lt;p_1/p_2$</span>, I get <span class="math-container">$(e,0)$</span> and <span class="math-container">$ (0,e \cdot p_1/p_2)$</span> pairs, but I don't know how to solve the case where
<span class="math-container">$A=p_1/p_2$</span>.</p>
<p>For the second agent, I got this from the maximization problem: <span class="math-container">$x_{21} = p_2 \cdot e /2 p_1$</span> and <span class="math-container">$x_{22} = e/2$</span>.</p>
<p>So the first question is how to find demand in case of <span class="math-container">$A=p_1/p_2$</span> and how to construct Edgeworth Box in that case.</p>

CC BY-SA 4.0
2022-12-14T18:44:37.607Z
There is no demand "function" because a function must have a unique value. In the case you mentioned, everything on the budget line is a utility maximizing bundle.
Michael Greinecker
CC BY-SA 4.0
2022-12-15T06:50:07.300Z
Thanks but when $A = p_1/ p_2$ what we get for $x_{11}$ and $x_{12}$ pair ? also, may I ask what do you mean by the uniqueness of the function?
Tatanik501
CC BY-SA 4.0
2022-12-15T07:25:56.473Z
A function from a set $A$ to a set $B$ associates with every point in the set $A$ exactly one point in the set $B$. Here, you don't get one point, you get many pairs. Namely, the whole budget line.
Michael Greinecker
CC BY-SA 4.0
2022-12-15T08:39:37.267Z
Thank you sir I understand it now.
Tatanik501
<microeconomics><general-equilibrium><demand><walrasian>
Finding demand function in Walrasian equilibrium
Tatanik501
# Finding demand function in Walrasian equilibrium

(Asked by: username_0 on 2022-12-14)

Maybe the title doesn't reflect what I mean perfectly but basically, I wanna derive demand functions from those two utility functions:
*where $x\_{11}$ is the consumption of good 1 by agent 1 and $x\_{12}$ is the consumption of good 2 by agent 1 and so on.*
$$
u\_1(x\_{11},x\_{12}) = A \cdot x\_{11}+x\_{12}
$$

and endowments of agent 1 are $e\_1 = (e,0)$

$$
u\_2(x\_{21},x\_{22}) = x\_{21}\cdot x\_{22}
$$

and endowments of agent 2 are $e\_2 = (0,e)$

When $A>p\_1/p\_2$ or $A, I get $(e,0)$ and $ (0,e \cdot p\_1/p\_2)$ pairs, but I don't know how to solve the case where
$A=p\_1/p\_2$.

For the second agent, I got this from the maximization problem: $x\_{21} = p\_2 \cdot e /2 p\_1$ and $x\_{22} = e/2$.

So the first question is how to find demand in case of $A=p\_1/p\_2$ and how to construct Edgeworth Box in that case.

**Tags:** microeconomics, general-equilibrium, demand, walrasian

> **Comments:**
> **username_1**: There is no demand "function" because a function must have a unique value. In the case you mentioned, everything on the budget line is a utility maximizing bundle.  
> **username_0**: Thanks but when $A = p_1/ p_2$ what we get for $x_{11}$ and $x_{12}$ pair ? also, may I ask what do you mean by the uniqueness of the function?  
> **username_1**: A function from a set $A$ to a set $B$ associates with every point in the set $A$ exactly one point in the set $B$. Here, you don't get one point, you get many pairs. Namely, the whole budget line.  
> **username_0**: Thank you sir I understand it now.  

---

## 2 answers

### Answer by username_2 on 2024-10-04. Score: 0

Given an exchange economy:

* Two consumers - $u\_1(x\_1,y\_1)=Ax\_1+y\_1$, where $A>0$ is given, $u\_2(x\_2,y\_2)=x\_2y\_2$.
* Endowments - $\omega\_1=(e,0)$, $\omega\_2=(0,e)$, where $e>0$ is given.

Set of feasible allocations is $\mathcal{F}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_1+x\_2=y\_1+y\_2=e\}$

Set of Pareto efficient allocations is

$\mathcal{PE}=\begin{cases} \{((x\_1,y\_1),(x\_2,y\_2))\in\mathcal{F}|y\_2=\min(Ax\_2,e)\} & \text{if } A \geq 1 \\ \{((x\_1,y\_1),(x\_2,y\_2))\in\mathcal{F}|(y\_2=Ax\_2) \ \vee \ (y\_2\geq Ae \ \wedge \ x\_2=e) \} & \text{if } A < 1\end{cases}$

Competitive equilibrium allocation and the supporting price is

$(((x\_1^c,y\_1^c),(x\_2^c,y\_2^c)),(p\_X^c,p\_Y^c))=\begin{cases} \left(\left(\left(\frac{(2A-1)e}{2A},\frac{e}{2}\right),\left(\frac{e}{2A},\frac{e}{2}\right)\right),(A,1) \right) & \text{if } A\geq\frac{1}{2} \\ \left(\left(\left(0,\frac{e}{2}\right),\left(e,\frac{e}{2}\right)\right),(\frac{1}{2},1) \right) & \text{if } A<\frac{1}{2} \end{cases}$


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### Answer by username_3 on 2024-10-03. Score: 0

We have the following information,

$u\_1(x\_{11},x\_{12})=A\cdot x\_{11}+x\_{12}$

$u\_2(x\_{11},x\_{12})=x\_{21}\cdot x\_{22}$

$e\_1=(e,0)$

$e\_2=(0,e)$

Solving Agent 1's utility maximization problem,
$$\begin{eqnarray}
\max\_{{(x\_{11},x\_{12}}) \ \in \ \mathbb{R}^2\_+} & A\cdot x\_{11}+x\_{12} \\
& s.t. p\_1x\_{11}+p\_2x\_{12}\leq p\_1e
\end{eqnarray}$$
where $p\_1e$ is the value of agent 1's endowment, we get the agent 1's demand correspondence as,
$$\begin{eqnarray}
(x\_{11}^d,x\_{12}^d)\in
\begin{cases}
\{(0,\frac{p\_1}{p\_2}e)\} & \text{if} \ A < \frac{p\_1}{p\_2}\\
\{(e,0)\} & \text{if} \ A > \frac{p\_1}{p\_2}\\
\{(x\_{11},x\_{12})\in \mathbb{R}^2\_+ \ | \ p\_1x\_{11}+p\_2x\_{12}=p\_1e\}
& \text{if} \ A = \frac{p\_1}{p\_2}\\
\end{cases}
\end{eqnarray}
$$
Solving Agent 2's utility maximization problem,
$$\begin{eqnarray}
\max\_{{(x\_{21},x\_{22}})\ \in \ \mathbb{R}^2\_+} & x\_{11} \cdot x\_{12} \\
& s.t. p\_1x\_{21}+p\_2x\_{22}\leq p\_2e
\end{eqnarray}$$
where $p\_2e$ is the value of agent 2's endowment, we get the agent 2's demand function as,
$$(x\_{21}^d,x\_{22}^d)=\left(\frac{p\_2e}{2p\_1},\frac{e}{2} \right)$$

let's take price of good 2 as the numeraire and then our demand functions take the form,
$$\begin{eqnarray}
(x\_{11}^d,x\_{12}^d)\in
\begin{cases}
\{(0,p\_1e)\} & \text{if} \ A < p\_1\\
\{(e,0)\} & \text{if} \ A > p\_1\\
\{(x\_{11},x\_{12})\in \mathbb{R}^2\_+ \ | \ p\_1x\_{11}+x\_{12}=p\_1e\}
& \text{if} \ A = p\_1\\
\end{cases}
\end{eqnarray}
$$
and
$$(x\_{21}^d,x\_{22}^d)=\left(\frac{e}{2p\_1},\frac{e}{2} \right)$$

Now we can solve for the competitive equilibrium of this economy, I will assume WLOG that $A>0$,
so competitive equilibrium price vector will be dependent on the different values that $A$ takes so I will consider three different cases for the same,

Case 1, $\ 0, Case 2, $\ A=1$,and Case 3, $\ A>1$

So we have the following equilibrium price vector,
$$
(p\_1,p\_2)=(A,1)
$$
Here is the example of how the Edgeworth Box would look like for the case when $A=\frac{p\_1}{p\_2} = 1$
[![enter image description here](https://i.sstatic.net/M6kzLjopm.png)](https://i.sstatic.net/M6kzLjopm.png)

Intuitively one can reason out that $(p\_1,p\_2)=(A,1)$ is the competitive equilibrium price vector since when $\frac{p\_1}{p\_2}=A$, agent 2 will maximise his utility and given the budget constraint and the remaining quantities will be given to agent 1, and for any other price vector other that this there will be excess demand or excess supply of either good in the economy.


---

