<p>For the good in question, let <span class="math-container">$d^m(p,m)$</span> denote the Marshallian demand and let <span class="math-container">$d^h(p,u)$</span> denote the Hicksian demand, where <span class="math-container">$p$</span> is the price vector, <span class="math-container">$m$</span> is income, and <span class="math-container">$u$</span> is utility. Since the good is inferior <span class="math-container">$d^m$</span> is decreasing in <span class="math-container">$m$</span>.</p>
<p>We have the identity</p>
<p><span class="math-container">$$d^h(p,u)=d^m\big(p,e(p,u)\big) \tag{X}$$</span></p>
<p>where <span class="math-container">$e$</span> is the (minimum) expenditure function. The expenditure function must be (weakly) increasing (i.e. nondecreasing) in <span class="math-container">$u$</span>. If not, then there exist <span class="math-container">$u_1$</span> and <span class="math-container">$u_2$</span> with <span class="math-container">$u_2&gt;u_1$</span> such that <span class="math-container">$e(p,u_2)&lt;e(p,u_1)$</span>. But this violates the definition of the expenditure function. So, an increase in <span class="math-container">$u$</span> increases <span class="math-container">$e(p,u)$</span>, and since the good is inferior, this decreases the RHS of (X), i.e. the Marhsallian demand. Thus the LHS of (X) (the Hicksian demand) is (weakly) decreasing in <span class="math-container">$u$</span>.</p>

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2023-11-11T12:14:47.790Z
2023-11-11T12:14:47.790Z
smcc
<p>How can you rigorously show that Hicksian demand for an inferior good will decrease when utility increases?</p>
<p>Thanks,</p>

CC BY-SA 4.0
2023-11-09T23:09:29.380Z
You make the assumptions that guarantee that Hicksian demand equals Marshallian demand at the expenditure function and for the expenditure function to be increasing in utility depending on whether you want strict decreases. The rest is the definition of an inferior good.
Michael Greinecker
<utility><consumer-theory><demand><giffen-goods>
Change in Hicksian Demand of an Inferior Good when changing Utility
Peter Luu
# Change in Hicksian Demand of an Inferior Good when changing Utility

(Asked by: username_0 on 2023-11-09)

How can you rigorously show that Hicksian demand for an inferior good will decrease when utility increases?

Thanks,

**Tags:** utility, consumer-theory, demand, giffen-goods

> **Comments:**
> **username_1**: You make the assumptions that guarantee that Hicksian demand equals Marshallian demand at the expenditure function and for the expenditure function to be increasing in utility depending on whether you want strict decreases. The rest is the definition of an inferior good.  

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## 1 answers

### Answer by username_2 on 2023-11-11. Score: 1

For the good in question, let $d^m(p,m)$ denote the Marshallian demand and let $d^h(p,u)$ denote the Hicksian demand, where $p$ is the price vector, $m$ is income, and $u$ is utility. Since the good is inferior $d^m$ is decreasing in $m$.

We have the identity

$$d^h(p,u)=d^m\big(p,e(p,u)\big) \tag{X}$$

where $e$ is the (minimum) expenditure function. The expenditure function must be (weakly) increasing (i.e. nondecreasing) in $u$. If not, then there exist $u\_1$ and $u\_2$ with $u\_2>u\_1$ such that $e(p,u\_2)<e(p,u\_1)$. But this violates the definition of the expenditure function. So, an increase in $u$ increases $e(p,u)$, and since the good is inferior, this decreases the RHS of (X), i.e. the Marhsallian demand. Thus the LHS of (X) (the Hicksian demand) is (weakly) decreasing in $u$.


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