<p>One possibility is to reparameterize your expenditure as follow: <span class="math-container">$$e(\mathbf{p},u) = E(f(\mathbf{p}),g(\mathbf{p})u),$$</span> with <span class="math-container">$f,g:\mathbb{R}^n\rightarrow \mathbb{R}$</span>. Then use the chain rule to find <span class="math-container">$\partial{e}/\partial{p_j}$</span>, <span class="math-container">$\partial{e}/\partial{u}$</span> and to prove that <span class="math-container">$e$</span> is homogeneous of degree one in <span class="math-container">$\mathbf{p}$</span> iff ...</p>

CC BY-SA 4.0
2020-12-06T10:47:43.590Z
2020-12-06T10:47:43.590Z
Bertrand
<p>I have to find parameters m and d for which expenditure function is non-decreasing and homogeneous of degree 1.
My expenditure function is:</p>
<p>I think that I should find ∂e/∂p which has to be &gt;= 0 but I have a problem with this because expenditure function is for n-good. Can someone help me?</p>

CC BY-SA 4.0
2020-12-05T15:03:56.863Z
Your expenditure function is a bit unclear. What is $z$? Also you can use latex to write the equation for clarity.
Dayne
CC BY-SA 4.0
2020-12-05T15:30:20.640Z
Z is a parameter. I changed names of this parameters so as not to cause confusion.
Azsb
CC BY-SA 4.0
2020-12-06T10:13:54.617Z
There are two ambiguities here: (i) this looks as an exercise... which we are not supposed to solve. (ii) it seems that you use the \textbf{same} notation $e$ for denoting two \textbf{different} things. This may prevent us to answer or to give hints.
Bertrand
CC BY-SA 4.0
2020-12-06T10:19:45.833Z
Second "e" in equation is exp.
Azsb
<utility><consumer-theory>
When is expenditure function non-decreasing?
Azsb
# When is expenditure function non-decreasing?

(Asked by: username_0 on 2020-12-05)

I have to find parameters m and d for which expenditure function is non-decreasing and homogeneous of degree 1.
My expenditure function is:

I think that I should find ∂e/∂p which has to be >= 0 but I have a problem with this because expenditure function is for n-good. Can someone help me?

**Tags:** utility, consumer-theory

> **Comments:**
> **username_1**: Your expenditure function is a bit unclear. What is $z$? Also you can use latex to write the equation for clarity.  
> **username_0**: Z is a parameter. I changed names of this parameters so as not to cause confusion.  
> **username_2**: There are two ambiguities here: (i) this looks as an exercise... which we are not supposed to solve. (ii) it seems that you use the \textbf{same} notation $e$ for denoting two \textbf{different} things. This may prevent us to answer or to give hints.  
> **username_0**: Second "e" in equation is exp.  

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## 1 answers

### Answer by username_2 on 2020-12-06. Score: 1

One possibility is to reparameterize your expenditure as follow: $$e(\mathbf{p},u) = E(f(\mathbf{p}),g(\mathbf{p})u),$$ with $f,g:\mathbb{R}^n\rightarrow \mathbb{R}$. Then use the chain rule to find $\partial{e}/\partial{p\_j}$, $\partial{e}/\partial{u}$ and to prove that $e$ is homogeneous of degree one in $\mathbf{p}$ iff ...


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