<p>Walrasian demand <span class="math-container">$(x_1^d,x_2^d)$</span> solves the following problem:</p>
<p><span class="math-container">\begin{eqnarray*} \max_{x_1\geq 0,x_2\geq 0} &amp; \min(2x_1+x_2,x_1+2x_2) \\ \text{s.t.} &amp; p_1x_1+p_2x_2\leq m\end{eqnarray*}</span>
where <span class="math-container">$p_1&gt;0$</span>, <span class="math-container">$p_2&gt;0$</span> and <span class="math-container">$m\geq 0$</span> are given.</p>
<p>Here is the solution and the indirect utility function:</p>
<p><span class="math-container">\begin{eqnarray*} (x_1^d,x_2^d)\in\begin{cases}  \left\{\left(\dfrac{m}{p_1},0\right)\right\} &amp; \text{if } \dfrac{p_1}{p_2} &lt;  \dfrac{1}{2} \\ \left\{\left(x_1,x_2\right)\in\mathbb{R}^2_+|p_1x_1+p_2x_2=m \ \wedge \ x_1\geq x_2\right\} &amp; \text{if } \dfrac{p_1}{p_2} =  \dfrac{1}{2} \\ \left\{\left(\dfrac{m}{p_1+p_2},\dfrac{m}{p_1+p_2}\right)\right\} &amp; \text{if }    \dfrac{1}{2}&lt; \dfrac{p_1}{p_2} &lt; 2 \\ \left\{\left(x_1,x_2\right)\in\mathbb{R}^2_+|p_1x_1+p_2x_2=m \ \wedge \ x_1\leq x_2\right\} &amp; \text{if } \dfrac{p_1}{p_2} =  2 \\ \left\{\left(0,\dfrac{m}{p_2}\right)\right\} &amp; \text{if } \dfrac{p_1}{p_2} &gt; 2\end{cases}\end{eqnarray*}</span></p>
<p>Indirect utility function is
<span class="math-container">\begin{eqnarray*} V(p_1,p_2,m) = \min(2x_1^d+x_2^d,x_1^d+2x_2^d) = \max\left(\dfrac{m}{p_1}, \dfrac{m}{p_2}, \dfrac{3m}{p_1+p_2}\right)\end{eqnarray*}</span></p>
<p><a href="https://i.sstatic.net/TRwGr.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TRwGr.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-09-13T11:35:17.977Z
2023-09-13T12:06:33.880Z
2023-09-13T12:06:33.880Z
Amit
<p>Given the utility <span class="math-container">$u = \min \{x2 + 2x1, x1 + 2x2\}$</span>, obtain the Walrasian Demand and the Indirect Utility Function.</p>

CC BY-SA 4.0
2023-09-12T20:28:04.990Z
You can refer to page 2 of this document for the Walrasian demand: http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/notes_1.pdf
Amit
<microeconomics>
Walrasian Demand and Indirect Utility Function
Víctor Urbina
# Walrasian Demand and Indirect Utility Function

(Asked by: username_0 on 2023-09-12)

Given the utility $u = \min \{x2 + 2x1, x1 + 2x2\}$, obtain the Walrasian Demand and the Indirect Utility Function.

**Tags:** microeconomics

> **Comments:**
> **username_1**: You can refer to page 2 of this document for the Walrasian demand: http://economicsentrance.weebly.com/uploads/1/1/0/5/1105777/notes_1.pdf  

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## 1 answers

### Answer by username_1 on 2023-09-13. Score: 1

Walrasian demand $(x\_1^d,x\_2^d)$ solves the following problem:

\begin{eqnarray\*} \max\_{x\_1\geq 0,x\_2\geq 0} & \min(2x\_1+x\_2,x\_1+2x\_2) \\ \text{s.t.} & p\_1x\_1+p\_2x\_2\leq m\end{eqnarray\*}
where $p\_1>0$, $p\_2>0$ and $m\geq 0$ are given.

Here is the solution and the indirect utility function:

\begin{eqnarray\*} (x\_1^d,x\_2^d)\in\begin{cases} \left\{\left(\dfrac{m}{p\_1},0\right)\right\} & \text{if } \dfrac{p\_1}{p\_2} < \dfrac{1}{2} \\ \left\{\left(x\_1,x\_2\right)\in\mathbb{R}^2\_+|p\_1x\_1+p\_2x\_2=m \ \wedge \ x\_1\geq x\_2\right\} & \text{if } \dfrac{p\_1}{p\_2} = \dfrac{1}{2} \\ \left\{\left(\dfrac{m}{p\_1+p\_2},\dfrac{m}{p\_1+p\_2}\right)\right\} & \text{if } \dfrac{1}{2}< \dfrac{p\_1}{p\_2} < 2 \\ \left\{\left(x\_1,x\_2\right)\in\mathbb{R}^2\_+|p\_1x\_1+p\_2x\_2=m \ \wedge \ x\_1\leq x\_2\right\} & \text{if } \dfrac{p\_1}{p\_2} = 2 \\ \left\{\left(0,\dfrac{m}{p\_2}\right)\right\} & \text{if } \dfrac{p\_1}{p\_2} > 2\end{cases}\end{eqnarray\*}

Indirect utility function is
\begin{eqnarray\*} V(p\_1,p\_2,m) = \min(2x\_1^d+x\_2^d,x\_1^d+2x\_2^d) = \max\left(\dfrac{m}{p\_1}, \dfrac{m}{p\_2}, \dfrac{3m}{p\_1+p\_2}\right)\end{eqnarray\*}

[![enter image description here](https://i.sstatic.net/TRwGr.png)](https://i.sstatic.net/TRwGr.png)


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