<p>Fix some good <span class="math-container">$r$</span>. For notational convenience, let me drop the supscript <span class="math-container">$h$</span>. From the first order conditions, we get:
<span class="math-container">$$
x_{j} = \zeta_j + \left(\frac{p_{r}}{p_{j}}\right)^{\sigma}(x_{r} - \zeta_{r})
$$</span>
Substitute into the budget constraint <span class="math-container">$\sum_{j} p_j x_j = w$</span>.
<span class="math-container">$$
\begin{align*}
&amp;\sum_j p_j \left(\frac{p_r}{p_j}\right)^\sigma (x_r - \zeta_r) + \sum_j p_j \zeta_j = w,\\
\iff &amp; \sum_j p_j^{1 - \sigma} p_r^\sigma (x_r - \zeta_r) = w - \sum_j p_j \zeta_j,\\
\iff &amp; p_r^\sigma (x_r - \zeta_r) = \frac{w - \sum_j p_j \zeta_j}{\sum_j p_j^{1 - \sigma}},\\
\iff &amp; x_r = \zeta_r +  \frac{w- \sum_j p_j \zeta_j}{p_r^\sigma \sum_j p_j^{1 - \sigma}}
\end{align*}
$$</span></p>
<p>Now substitute this into the utility function
<span class="math-container">$$\left(\sum_r (x_r - \zeta_r)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}.
$$</span></p>
<p>We obtain:
<span class="math-container">$$
\begin{align*}
&amp;\left(\sum_r \left(\frac{w - \sum_j p_j \zeta_j}{p_r^\sigma \sum_j p_j^{1 - \sigma}}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}},\\
=&amp; \left[w - \sum_j p_j \zeta_j\right] \frac{\left(\sum_r p_r^{1 - \sigma}\right)^{\frac{\sigma}{\sigma - 1}}}{\sum_j p_j^{1 - \sigma}},\\
= &amp; \frac{\left[w - \sum_j p_j \zeta_j\right]}{\left(\sum_r p_r^{1 - \sigma}\right)^{\frac{1}{1-\sigma}}},\\
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-04-08T06:53:22.817Z
2024-04-08T06:53:22.817Z
tdm
<p>Consider the following CES utility function for the <span class="math-container">$h$</span> household
<span class="math-container">\begin{equation}
U^h(x^h_1,\ldots,x^h_N) = \left[ \sum_{j=1}^{N} (x^h_j - \zeta^h_j)^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}}
\end{equation}</span>
where <span class="math-container">$\sigma \in (0,\infty)$</span> denotes the elasticity of substitution and <span class="math-container">$\zeta^h_j \in [-\bar{\zeta}, \bar{\zeta}]$</span> denotes whether the particular good is a necessity or not for the household. For example, <span class="math-container">$\zeta^h_j &gt;0$</span> may mean that
household h needs to consume at least a certain amount of good <span class="math-container">$j$</span> to survive.</p>
<p>It is assumed that</p>
<p><span class="math-container">\begin{equation}
\sum_{j=1}^{N} p_j \bar{\zeta} &lt; w^h
\end{equation}</span>
where <span class="math-container">$w^h$</span> denotes the household <span class="math-container">$w^h$</span> income.</p>
<p>Now, to derive the indirect utility function, that according to the book reads
<span class="math-container">\begin{equation}
\nu^h(p,w^h)= \frac{\left[- \sum_{j=1}^{N} p_j \zeta^h_j + w^h \right]}{\left[\sum_{j=1}^{N} p^{1-\sigma}_j \right]^{\frac{1}{1-\sigma}}}
\end{equation}</span></p>
<p>I should find the optimal consumption bundles and then plug them into the utility function. I proceeded as follows</p>
<p>Considering two commodities <span class="math-container">$x^h_j$</span> and <span class="math-container">$x^h_{j'}$</span> with <span class="math-container">$j \ne j'$</span>:
<span class="math-container">\begin{equation}
\frac{\partial \mathcal{L}}{\partial x_j} \Bigg/ \frac{\partial \mathcal{L}}{\partial x_j'} =\left( \frac{{x^h_{j'}-\zeta^h_{j'}}}{{x^h_{j}-\zeta^h_{j}}}\right)^{\frac{1}{\sigma}} = \frac{p_j}{p_{j'}}
\end{equation}</span>
where <span class="math-container">$\mathcal{L}$</span> denotes the lagrangian. The constraint I used is: <span class="math-container">$\sum_{j=1}^{N} p_j x_j = w^h$</span>.
Now, solving the expression above by <span class="math-container">$x^h_{j'}$</span> and plugging it into the constraint
<span class="math-container">\begin{equation}
\sum_{j \ne j'} p_j x^h_j + (p_j)^{\sigma} p^{1-\sigma}_{j'} (x^h_j - \zeta^h_j) +  p_{j'} \zeta^h_{j'}=w^h
\end{equation}</span>
I'm not sure that what I'm doing is correct. If so, how should I continue?</p>

<microeconomics><utility>
Derive indirect utility function - Problem with CES
Maximilian
# Derive indirect utility function - Problem with CES

(Asked by: username_0 on 2024-03-30)

Consider the following CES utility function for the $h$ household
\begin{equation}
U^h(x^h\_1,\ldots,x^h\_N) = \left[ \sum\_{j=1}^{N} (x^h\_j - \zeta^h\_j)^{\frac{\sigma-1}{\sigma}} \right]^{\frac{\sigma}{\sigma-1}}
\end{equation}
where $\sigma \in (0,\infty)$ denotes the elasticity of substitution and $\zeta^h\_j \in [-\bar{\zeta}, \bar{\zeta}]$ denotes whether the particular good is a necessity or not for the household. For example, $\zeta^h\_j >0$ may mean that
household h needs to consume at least a certain amount of good $j$ to survive.

It is assumed that

\begin{equation}
\sum\_{j=1}^{N} p\_j \bar{\zeta} < w^h
\end{equation}
where $w^h$ denotes the household $w^h$ income.

Now, to derive the indirect utility function, that according to the book reads
\begin{equation}
\nu^h(p,w^h)= \frac{\left[- \sum\_{j=1}^{N} p\_j \zeta^h\_j + w^h \right]}{\left[\sum\_{j=1}^{N} p^{1-\sigma}\_j \right]^{\frac{1}{1-\sigma}}}
\end{equation}

I should find the optimal consumption bundles and then plug them into the utility function. I proceeded as follows

Considering two commodities $x^h\_j$ and $x^h\_{j'}$ with $j \ne j'$:
\begin{equation}
\frac{\partial \mathcal{L}}{\partial x\_j} \Bigg/ \frac{\partial \mathcal{L}}{\partial x\_j'} =\left( \frac{{x^h\_{j'}-\zeta^h\_{j'}}}{{x^h\_{j}-\zeta^h\_{j}}}\right)^{\frac{1}{\sigma}} = \frac{p\_j}{p\_{j'}}
\end{equation}
where $\mathcal{L}$ denotes the lagrangian. The constraint I used is: $\sum\_{j=1}^{N} p\_j x\_j = w^h$.
Now, solving the expression above by $x^h\_{j'}$ and plugging it into the constraint
\begin{equation}
\sum\_{j \ne j'} p\_j x^h\_j + (p\_j)^{\sigma} p^{1-\sigma}\_{j'} (x^h\_j - \zeta^h\_j) + p\_{j'} \zeta^h\_{j'}=w^h
\end{equation}
I'm not sure that what I'm doing is correct. If so, how should I continue?

**Tags:** microeconomics, utility


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## 1 answers

### Answer by username_1 on 2024-04-08. Score: 1 | Accepted answer

Fix some good $r$. For notational convenience, let me drop the supscript $h$. From the first order conditions, we get:
$$
x\_{j} = \zeta\_j + \left(\frac{p\_{r}}{p\_{j}}\right)^{\sigma}(x\_{r} - \zeta\_{r})
$$
Substitute into the budget constraint $\sum\_{j} p\_j x\_j = w$.
$$
\begin{align\*}
&\sum\_j p\_j \left(\frac{p\_r}{p\_j}\right)^\sigma (x\_r - \zeta\_r) + \sum\_j p\_j \zeta\_j = w,\\
\iff & \sum\_j p\_j^{1 - \sigma} p\_r^\sigma (x\_r - \zeta\_r) = w - \sum\_j p\_j \zeta\_j,\\
\iff & p\_r^\sigma (x\_r - \zeta\_r) = \frac{w - \sum\_j p\_j \zeta\_j}{\sum\_j p\_j^{1 - \sigma}},\\
\iff & x\_r = \zeta\_r + \frac{w- \sum\_j p\_j \zeta\_j}{p\_r^\sigma \sum\_j p\_j^{1 - \sigma}}
\end{align\*}
$$

Now substitute this into the utility function
$$\left(\sum\_r (x\_r - \zeta\_r)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}}.
$$

We obtain:
$$
\begin{align\*}
&\left(\sum\_r \left(\frac{w - \sum\_j p\_j \zeta\_j}{p\_r^\sigma \sum\_j p\_j^{1 - \sigma}}\right)^{\frac{\sigma - 1}{\sigma}}\right)^{\frac{\sigma}{\sigma - 1}},\\
=& \left[w - \sum\_j p\_j \zeta\_j\right] \frac{\left(\sum\_r p\_r^{1 - \sigma}\right)^{\frac{\sigma}{\sigma - 1}}}{\sum\_j p\_j^{1 - \sigma}},\\
= & \frac{\left[w - \sum\_j p\_j \zeta\_j\right]}{\left(\sum\_r p\_r^{1 - \sigma}\right)^{\frac{1}{1-\sigma}}},\\
\end{align\*}
$$


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