<p>Take the following utility function:
<span class="math-container">$$
\prod_{s \in S} \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{s}}
$$</span></p>
<p>Which is a CES nested in a Cobb-Douglas.</p>
<p>Consider the sub-utility function:
<span class="math-container">$$
Q_{s} = \left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}}.
$$</span>
Then we can re-write the utility function succinctly as:
<span class="math-container">$$
U = \prod_{s \in S} Q_s^{\mu_s}
$$</span>
To solve the utility maximisation problem. Notice that <span class="math-container">$U$</span> is separable in the subgroups. As such, we can solve the problem using two stage budgeting.</p>
<ol>
<li>In the first step, we can determine the optimal allocation within each subgroup <span class="math-container">$s$</span> by maximize the sub-utility functions <span class="math-container">$Q_s$</span> given the total expenditure <span class="math-container">$E_s$</span> given on each subgroup <span class="math-container">$s$</span>.</li>
<li>In the second step, we determine the optimal allocation of income across each subgroup <span class="math-container">$s$</span>.</li>
</ol>
<p>For Step 1, we solve the following problem:
<span class="math-container">$$
\max\left(\sum_{i \in N} x_{i,s}^{\frac{\sigma_s - 1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s - 1}} \text{subject to } \sum_{i \in N} p_{i,s} x_{i,s} = E_s.
$$</span>
This is a CES utility function. The first order condition gives:
<span class="math-container">$$
Q_s \frac{x_{i,s}^{-\frac{1}{\sigma_s}}}{\sum_{w \in N} x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}} = \lambda p_{i,s}.
$$</span>
Then:
<span class="math-container">$$
x_{i,s} = \lambda^{-\sigma_s} p_{i,s}^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-1\sigma_s}
$$</span>
Multiplying by <span class="math-container">$p_{i,s}$</span> and adding over all <span class="math-container">$i \in N$</span> gives:
<span class="math-container">$$
\begin{align*}
E_s = \sum_{i \in N} p_{i,s} x_{i,s} &amp;= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} \sum_{i \in N} p_{i,s}^{1 - \sigma_s}\\
&amp;= \lambda^{-\sigma_s} \left(\frac{\sum_w x_{w,s}^{\frac{\sigma_s - 1}{\sigma_s}}}{Q_s}\right)^{-\sigma_s} P_s^{1 - \sigma_s}
\end{align*}
$$</span>
Where <span class="math-container">$E_{s} = \sum_{i\in N} p_{i,s} x_{i,s}$</span> and <span class="math-container">$P_{s}^{1- \sigma_s} = \sum_{i \in N} p_{i,s}^{1- \sigma_s}$</span>.</p>
<p>Then:
<span class="math-container">$$
x_{i,s} = E_{s} \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1-\sigma_s}} \tag{1}
$$</span>
Let's compute the value of <span class="math-container">$Q_s$</span>:
<span class="math-container">$$
\begin{align*}
&amp;x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1 -\sigma_s}}\right)^{\frac{\sigma_s-1}{ \sigma_s}},\\
\to &amp;\sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}= E_{j,s}^{\frac{\sigma_s-1}{\sigma_s}} \left(\frac{1}{P_{s}^{1- \sigma_s}}\right)^{\frac{\sigma_s-1}{\sigma_s}} \sum_{i \in N} p_{i,s}^{(1-\sigma_s)},\\
\to&amp; \sum_{i \in N} x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}} = E_{s}^{\frac{\sigma_s-1}{\sigma_s}} P_{s}^{\frac{1 - \sigma_s}{\sigma_s}}\\
\to &amp; \left(\sum_i x_{i,s}^{\frac{\sigma_s-1}{\sigma_s}}\right)^{\frac{\sigma_s}{\sigma_s-1}} = E_{s} P_{s}^{-1}\\
\to&amp; P_{s} Q_s = E_{s}.
\end{align*}
$$</span>
This shows that the price index <span class="math-container">$P_s$</span> and the quantity index <span class="math-container">$Q_s$</span> are exact price and quanty indices for this subgroup (as is expected as we were maximising a CES utility function).</p>
<p>Let's now go to step 2. Notice that the budget constraint can be rewritten as:
<span class="math-container">$$
E = \sum_{s} E_s = \sum_s P_s Q_s.
$$</span>
We want to determine <span class="math-container">$E_s$</span> to maximize:
<span class="math-container">$$
U = \prod_{j \in S} Q_{s}^{\mu_{j,s}} \text{ s.t. } \sum_s P_{s} Q_{s} = E.
$$</span>
This is a simple Cobb-Douglas utility function. As such, we know that expenditure shares are proportional to income:
<span class="math-container">$$
P_s Q_s = E_s = \frac{\mu_s}{\sum_w \mu_w} E
$$</span>
Finally, substitute this into the equation <span class="math-container">$(1)$</span>:
<span class="math-container">$$
x_{i,s} = \frac{\mu_{s}}{\sum_w \mu_w} E \frac{p_{i,s}^{-\sigma_s}}{P_{s}^{1- \sigma_s}}
$$</span></p>

CC BY-SA 4.0
2021-05-18T08:02:09.143Z
2021-05-18T11:17:22.767Z
2021-05-18T11:17:22.767Z
tdm
<p>I'm trying to understand the following paper: Hsieh &amp; Ossa: A global view of productivity growth in China (2016). The pdf can be found here: <a href="https://faculty.chicagobooth.edu/chang-tai.hsieh/research/hsieh_ossa_jie.pdf" rel="noreferrer">https://faculty.chicagobooth.edu/chang-tai.hsieh/research/hsieh_ossa_jie.pdf</a></p>

<p>I struggle to understand the utility maximization problem right at the beginning of the paper. Can someone show me how to maximize the following utility with respect to <span class="math-container">$x_{ijs}$</span>:</p>

<p><span class="math-container">$U_j=\prod_s^S\left(\sum_i^N \int_0^{M_is^e}x_{ijs}(\nu_{is})^{\frac{\sigma_s-1}{\sigma_s}}d\nu_{is} \right)^{\frac{\sigma_s}{\sigma_s-1}\mu_{js}}$</span></p>

<p>The end result is:</p>

<p><span class="math-container">$x_{ijs}=\frac{p_{ijs}^{-\sigma_s}}{P_{js}^{1-\sigma_s}}\mu_{js}E_j$</span></p>

<p>where <span class="math-container">$P_{js}=\left(\sum_i^N M_{is}^ep_{ijs}^{1-\sigma_s}\right)^\frac{1}{1-\sigma_s}$</span></p>

<p><span class="math-container">$N$</span>: number of countries </p>

<p><span class="math-container">$S$</span>: number of industries </p>

<p><span class="math-container">$M_{is}^e$</span>: number of entrants in industry s of country i</p>

<p><span class="math-container">$x_{ijs}$</span>: quantity of an industry s variety from country i consumed in country j</p>

<p><span class="math-container">$\mu_{js}$</span>: fraction of country j income spent on industry s varieties</p>

<p><span class="math-container">$\sigma_s&gt;1$</span>: elasticity of substitution between industry s varieties</p>

<p><span class="math-container">$p_{ijs}$</span>: price of an industry s variety from country i in country j</p>

<p><span class="math-container">$P_{js}$</span>: ideal price index in industry s of country j</p>

<p><span class="math-container">$E_j$</span>: total expenditure in country j</p>

CC BY-SA 4.0
2019-08-22T00:05:33.020Z
I dont understand why is there an integral in the utility function. I see that the amount of output purchased by j's consumers from the s-th sector in i ($x_{ijs}(ν_{is})$) depends on the number of entrants in s in i. But I dont understand why do they integrate all the values if$x_{ijs}$ from $x_{ijs}(0)$ to $x_{ijs}(M^e_{is})$, since they only consume $x_{ijs}(M^e_{is})$. Could you explain?
Grada Gukovic
<microeconomics><utility>
utility maximization with nested Cobb–Douglas–CES preferences
strizzle
# utility maximization with nested Cobb–Douglas–CES preferences

(Asked by: username_0 on 2019-08-15)

I'm trying to understand the following paper: Hsieh & Ossa: A global view of productivity growth in China (2016). The pdf can be found here: <https://faculty.chicagobooth.edu/chang-tai.hsieh/research/hsieh_ossa_jie.pdf>

I struggle to understand the utility maximization problem right at the beginning of the paper. Can someone show me how to maximize the following utility with respect to $x\_{ijs}$:

$U\_j=\prod\_s^S\left(\sum\_i^N \int\_0^{M\_is^e}x\_{ijs}(\nu\_{is})^{\frac{\sigma\_s-1}{\sigma\_s}}d\nu\_{is} \right)^{\frac{\sigma\_s}{\sigma\_s-1}\mu\_{js}}$

The end result is:

$x\_{ijs}=\frac{p\_{ijs}^{-\sigma\_s}}{P\_{js}^{1-\sigma\_s}}\mu\_{js}E\_j$

where $P\_{js}=\left(\sum\_i^N M\_{is}^ep\_{ijs}^{1-\sigma\_s}\right)^\frac{1}{1-\sigma\_s}$

$N$: number of countries

$S$: number of industries

$M\_{is}^e$: number of entrants in industry s of country i

$x\_{ijs}$: quantity of an industry s variety from country i consumed in country j

$\mu\_{js}$: fraction of country j income spent on industry s varieties

$\sigma\_s>1$: elasticity of substitution between industry s varieties

$p\_{ijs}$: price of an industry s variety from country i in country j

$P\_{js}$: ideal price index in industry s of country j

$E\_j$: total expenditure in country j

**Tags:** microeconomics, utility
> **Comments:**
> **username_1**: I dont understand why is there an integral in the utility function. I see that the amount of output purchased by j's consumers from the s-th sector in i ($x_{ijs}(ν_{is})$) depends on the number of entrants in s in i. But I dont understand why do they integrate all the values if$x_{ijs}$ from $x_{ijs}(0)$ to $x_{ijs}(M^e_{is})$, since they only consume $x_{ijs}(M^e_{is})$. Could you explain?  

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## 1 answers

### Answer by username_2 on 2021-05-18. Score: 4

Take the following utility function:
$$
\prod\_{s \in S} \left(\sum\_{i \in N} x\_{i,s}^{\frac{\sigma\_s-1}{\sigma\_s}} \right)^{\frac{\sigma\_s}{\sigma\_s-1}\mu\_{s}}
$$

Which is a CES nested in a Cobb-Douglas.

Consider the sub-utility function:
$$
Q\_{s} = \left(\sum\_{i \in N} x\_{i,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}\right)^{\frac{\sigma\_s}{\sigma\_s - 1}}.
$$
Then we can re-write the utility function succinctly as:
$$
U = \prod\_{s \in S} Q\_s^{\mu\_s}
$$
To solve the utility maximisation problem. Notice that $U$ is separable in the subgroups. As such, we can solve the problem using two stage budgeting.

1. In the first step, we can determine the optimal allocation within each subgroup $s$ by maximize the sub-utility functions $Q\_s$ given the total expenditure $E\_s$ given on each subgroup $s$.
2. In the second step, we determine the optimal allocation of income across each subgroup $s$.

For Step 1, we solve the following problem:
$$
\max\left(\sum\_{i \in N} x\_{i,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}\right)^{\frac{\sigma\_s}{\sigma\_s - 1}} \text{subject to } \sum\_{i \in N} p\_{i,s} x\_{i,s} = E\_s.
$$
This is a CES utility function. The first order condition gives:
$$
Q\_s \frac{x\_{i,s}^{-\frac{1}{\sigma\_s}}}{\sum\_{w \in N} x\_{w,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}} = \lambda p\_{i,s}.
$$
Then:
$$
x\_{i,s} = \lambda^{-\sigma\_s} p\_{i,s}^{-\sigma\_s} \left(\frac{\sum\_w x\_{w,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}}{Q\_s}\right)^{-1\sigma\_s}
$$
Multiplying by $p\_{i,s}$ and adding over all $i \in N$ gives:
$$
\begin{align\*}
E\_s = \sum\_{i \in N} p\_{i,s} x\_{i,s} &= \lambda^{-\sigma\_s} \left(\frac{\sum\_w x\_{w,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}}{Q\_s}\right)^{-\sigma\_s} \sum\_{i \in N} p\_{i,s}^{1 - \sigma\_s}\\
&= \lambda^{-\sigma\_s} \left(\frac{\sum\_w x\_{w,s}^{\frac{\sigma\_s - 1}{\sigma\_s}}}{Q\_s}\right)^{-\sigma\_s} P\_s^{1 - \sigma\_s}
\end{align\*}
$$
Where $E\_{s} = \sum\_{i\in N} p\_{i,s} x\_{i,s}$ and $P\_{s}^{1- \sigma\_s} = \sum\_{i \in N} p\_{i,s}^{1- \sigma\_s}$.

Then:
$$
x\_{i,s} = E\_{s} \frac{p\_{i,s}^{-\sigma\_s}}{P\_{s}^{1-\sigma\_s}} \tag{1}
$$
Let's compute the value of $Q\_s$:
$$
\begin{align\*}
&x\_{i,s}^{\frac{\sigma\_s-1}{\sigma\_s}} = E\_{s}^{\frac{\sigma\_s-1}{\sigma\_s}} \left(\frac{p\_{i,s}^{-\sigma\_s}}{P\_{s}^{1 -\sigma\_s}}\right)^{\frac{\sigma\_s-1}{ \sigma\_s}},\\
\to &\sum\_{i \in N} x\_{i,s}^{\frac{\sigma\_s-1}{\sigma\_s}}= E\_{j,s}^{\frac{\sigma\_s-1}{\sigma\_s}} \left(\frac{1}{P\_{s}^{1- \sigma\_s}}\right)^{\frac{\sigma\_s-1}{\sigma\_s}} \sum\_{i \in N} p\_{i,s}^{(1-\sigma\_s)},\\
\to& \sum\_{i \in N} x\_{i,s}^{\frac{\sigma\_s-1}{\sigma\_s}} = E\_{s}^{\frac{\sigma\_s-1}{\sigma\_s}} P\_{s}^{\frac{1 - \sigma\_s}{\sigma\_s}}\\
\to & \left(\sum\_i x\_{i,s}^{\frac{\sigma\_s-1}{\sigma\_s}}\right)^{\frac{\sigma\_s}{\sigma\_s-1}} = E\_{s} P\_{s}^{-1}\\
\to& P\_{s} Q\_s = E\_{s}.
\end{align\*}
$$
This shows that the price index $P\_s$ and the quantity index $Q\_s$ are exact price and quanty indices for this subgroup (as is expected as we were maximising a CES utility function).

Let's now go to step 2. Notice that the budget constraint can be rewritten as:
$$
E = \sum\_{s} E\_s = \sum\_s P\_s Q\_s.
$$
We want to determine $E\_s$ to maximize:
$$
U = \prod\_{j \in S} Q\_{s}^{\mu\_{j,s}} \text{ s.t. } \sum\_s P\_{s} Q\_{s} = E.
$$
This is a simple Cobb-Douglas utility function. As such, we know that expenditure shares are proportional to income:
$$
P\_s Q\_s = E\_s = \frac{\mu\_s}{\sum\_w \mu\_w} E
$$
Finally, substitute this into the equation $(1)$:
$$
x\_{i,s} = \frac{\mu\_{s}}{\sum\_w \mu\_w} E \frac{p\_{i,s}^{-\sigma\_s}}{P\_{s}^{1- \sigma\_s}}
$$


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