<p>Given indirect utility function <span class="math-container">$v(p,w)$</span>, where p are prices, w is wealth, my attempt is as following:</p>
<p>From the Utility Maximization Problem (UMP): Set up Lagrangian:</p>
<p><span class="math-container">$L = u(x) + \lambda[w - px]$</span></p>
<p><span class="math-container">$L_w = \lambda &gt; 0$</span>, by local non-satiation</p>
<p><span class="math-container">$L_p = -\lambda x  \leq 0$</span> since <span class="math-container">$x\geq 0$</span> and using <span class="math-container">$\lambda &gt; 0$</span></p>
<p>Is this ok?</p>

CC BY-SA 4.0
2022-03-15T08:11:11.040Z
Yes the argumentation is good, but the relationship between $L$ and $v$ is missing. I would approach this using the optimal values of $x$ and $\lambda$ and use the envelope theorem.
Bertrand
CC BY-SA 4.0
2022-03-15T09:56:33.207Z
The partial of indirect utility function wrt to these variables should be the same as the partial wrt to the Lagrangian function because they are continuously differentiable?
Kinno
<microeconomics>
Show indirect utility is increasing w and non-increasing in price?
Kinno
# Show indirect utility is increasing w and non-increasing in price?

(Asked by: username_0 on 2022-03-15)

Given indirect utility function $v(p,w)$, where p are prices, w is wealth, my attempt is as following:

From the Utility Maximization Problem (UMP): Set up Lagrangian:

$L = u(x) + \lambda[w - px]$

$L\_w = \lambda > 0$, by local non-satiation

$L\_p = -\lambda x \leq 0$ since $x\geq 0$ and using $\lambda > 0$

Is this ok?

**Tags:** microeconomics

> **Comments:**
> **username_1**: Yes the argumentation is good, but the relationship between $L$ and $v$ is missing. I would approach this using the optimal values of $x$ and $\lambda$ and use the envelope theorem.  
> **username_0**: The partial of indirect utility function wrt to these variables should be the same as the partial wrt to the Lagrangian function because they are continuously differentiable?  

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## 0 answers

