<p>A good way to think about this is to think about the cases where there are a finite number of goods, indexed by <span class="math-container">$\omega \in \Omega$</span>. In this case, the partial derivative is
<span class="math-container">$$
p(\omega) = \frac{\partial}{\partial c(\omega)} \sum_{\omega' \in \Omega} p(\omega') c(\omega').
$$</span>
The case in which <span class="math-container">$\Omega$</span> is a continuum of goods follows as a sort of limiting extension. See that even this finite sum can be expressed as an integral, given the proper measure <span class="math-container">$\mu$</span>:
<span class="math-container">$$
\sum_{\omega \in \Omega} p(\omega) c(\omega) = \int_{\Omega} p(\omega) c(\omega) \,\mu(\mathrm d \omega).
$$</span></p>

<p>You mention,</p>

<blockquote>
  <p>This is puzzling to me since the derivative of a indefinite integral
  of <span class="math-container">$c_t$</span> should be <span class="math-container">$c_t$</span>, not 1.</p>
</blockquote>

<p>If you provided your reasoning for this, it would be easier to see where the confusion is. In any case, hope this helps! </p>

<p>Also, this guide may provide further assistance: <a href="http://www.columbia.edu/~jid2106/td/dixitstiglitzbasics.pdf" rel="nofollow noreferrer">The basics of “Dixit-Stiglitz lite”</a></p>

CC BY-SA 4.0
2019-06-08T23:36:52.537Z
2019-06-10T01:40:14.283Z
2019-06-10T01:40:14.283Z
jmbejara
<p>In the solution to the problem below provided by my professor he applies a first order condition to the lagrangian for c<sub>t</sub>, getting p<sub>t</sub> for the partial derivative of the argument "<span class="math-container">$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$</span>". This is puzzling to me since the derivative of a indefinite integral of c<sub>t</sub> should be c<sub>t</sub> not 1. </p>

<p>What's going on with this derivative and is it correct?</p>

<p><span class="math-container">$\frac{d}{dc_t(t)} \int p_t(w)*c_t(w)dw$</span> = <span class="math-container">$p_t(w)$</span></p>

<p><a href="https://i.sstatic.net/96hZU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/96hZU.png" alt="Problem description"></a></p>

<p>Edit: Removed hand written notes.</p>

CC BY-SA 4.0
2019-06-08T18:53:20.020Z
Please don't post handwritten mathematical formulas as pictures. Use [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) instead.
Herr K.
<macroeconomics>
Dixit Stiglitz price index FOC?
lleontan
# Dixit Stiglitz price index FOC?

(Asked by: username_0 on 2019-06-07)

In the solution to the problem below provided by my professor he applies a first order condition to the lagrangian for ct, getting pt for the partial derivative of the argument "$\frac{d}{dc\_t(t)} \int p\_t(w)\*c\_t(w)dw$". This is puzzling to me since the derivative of a indefinite integral of ct should be ct not 1.

What's going on with this derivative and is it correct?

$\frac{d}{dc\_t(t)} \int p\_t(w)\*c\_t(w)dw$ = $p\_t(w)$

[![Problem description](https://i.sstatic.net/96hZU.png)](https://i.sstatic.net/96hZU.png)

Edit: Removed hand written notes.

**Tags:** macroeconomics
> **Comments:**
> **username_1**: Please don't post handwritten mathematical formulas as pictures. Use [MathJax](https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference) instead.  

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## 1 answers

### Answer by username_2 on 2019-06-08. Score: 1

A good way to think about this is to think about the cases where there are a finite number of goods, indexed by $\omega \in \Omega$. In this case, the partial derivative is
$$
p(\omega) = \frac{\partial}{\partial c(\omega)} \sum\_{\omega' \in \Omega} p(\omega') c(\omega').
$$
The case in which $\Omega$ is a continuum of goods follows as a sort of limiting extension. See that even this finite sum can be expressed as an integral, given the proper measure $\mu$:
$$
\sum\_{\omega \in \Omega} p(\omega) c(\omega) = \int\_{\Omega} p(\omega) c(\omega) \,\mu(\mathrm d \omega).
$$

You mention,

> This is puzzling to me since the derivative of a indefinite integral
> of $c\_t$ should be $c\_t$, not 1.

If you provided your reasoning for this, it would be easier to see where the confusion is. In any case, hope this helps!

Also, this guide may provide further assistance: [The basics of “Dixit-Stiglitz lite”](http://www.columbia.edu/~jid2106/td/dixitstiglitzbasics.pdf)


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