<p>I'm struggling with a problem where I want to derive the Hicksian demand from a given expenditure function <span class="math-container">$e(p,u)$</span>. I would do that by using Shephard's Lemma, so that the first partial derivative of <span class="math-container">$e$</span> with respect to <span class="math-container">$p_i$</span> gives the Hicksian demand for good <span class="math-container">$i$</span>. My problem is, however, that the expenditure function given does only depend on prices, not on <span class="math-container">$u$</span> (which does not make any sense to me since <span class="math-container">$e$</span> should minimize the costs given a certain utility level, so it is clearly dependent on <span class="math-container">$u$</span>). The function is given as <span class="math-container">$e(p,u) = p_1^{1/4}*(-p_2^{1/3})$</span>. Checking on the homogeneity of <span class="math-container">$e$</span> in prices, also does not lead me to a homogeneity of degree <span class="math-container">$1$</span> which contradicts the properties of the expenditure function. Am I overlooking something important? I would appreciate any help!</p>

CC BY-SA 4.0
2022-12-11T11:36:40.857Z
There is obviously a problem in your expenditure function. This is not an expenditure function for a rational consumer, and not sur that her demand function can be obtained by Shephard's lemma in this context.
Bertrand
CC BY-SA 4.0
2022-12-12T22:29:22.677Z
Is there possibly something else in the problem that could allow you to convert the form of the expenditure function to something containing utility?
H Huang
CC BY-SA 4.0
2022-12-13T07:23:46.077Z
Thank you! I only know that for e: D -> R, where D = {(p, u)|p,u >}
papagena
<microeconomics><utility><demand>
Expenditure function only dependent on prices
papagena
# Expenditure function only dependent on prices

(Asked by: username_0 on 2022-12-11)

I'm struggling with a problem where I want to derive the Hicksian demand from a given expenditure function $e(p,u)$. I would do that by using Shephard's Lemma, so that the first partial derivative of $e$ with respect to $p\_i$ gives the Hicksian demand for good $i$. My problem is, however, that the expenditure function given does only depend on prices, not on $u$ (which does not make any sense to me since $e$ should minimize the costs given a certain utility level, so it is clearly dependent on $u$). The function is given as $e(p,u) = p\_1^{1/4}\*(-p\_2^{1/3})$. Checking on the homogeneity of $e$ in prices, also does not lead me to a homogeneity of degree $1$ which contradicts the properties of the expenditure function. Am I overlooking something important? I would appreciate any help!

**Tags:** microeconomics, utility, demand

> **Comments:**
> **username_1**: There is obviously a problem in your expenditure function. This is not an expenditure function for a rational consumer, and not sur that her demand function can be obtained by Shephard's lemma in this context.  
> **username_2**: Is there possibly something else in the problem that could allow you to convert the form of the expenditure function to something containing utility?  
> **username_0**: Thank you! I only know that for e: D -> R, where D = {(p, u)|p,u >}  

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## 0 answers

