<p>To solve the problem you also need the equation:</p>

<p>$Y_t=\left(\int_0^1 Y_t(j)^{1-\nu} \ dj \right)^{\frac{1}{1-\nu}}$</p>

<p>Now, the FOC is the same for every $Y_t(j)$, so we only have to differentiate once. Applying the chain rule we get:</p>

<p>$\frac{\partial \Pi_t}{\partial Y_t(j)}=P_t \left(\int_0^1 Y_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}} Y_t(j)^{-\nu}-P_t(j)=0$</p>

<p>At this point, the trick of the trade is that  </p>

<p>$\left(\int_0^1 Y_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}}$ which is a part of the FOC, equals $Y_t^{\nu}$ (check the first equation)</p>

<p>Setting 
$\left(\int_0^1 Y_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}}= Y_t^{\nu}$ in the FOC, and solving for $Y_t(j)$ will give the desired result.</p>

CC BY-SA 3.0
2016-09-20T21:05:07.443Z
2016-09-20T21:05:07.443Z
Chris tie
<p>So I've seen this sort of thing where you basically differentiate with respect to each Y_j so the integral drops out (seen this in moral hazard chapter from 
MWG and Romer's endogenous growth model). </p>

<p>The foc becomes:</p>

<p>$P_t(Y_t(j)^{(1-v)})^{v/(1-v)}Y^{-v}-P_t(j)=0$</p>

<p>This gets the result. </p>

CC BY-SA 3.0
2016-08-18T17:03:42.477Z
2016-08-18T17:03:42.477Z
VCG
<p>I'm self-studying Herbst-Schorfheide book on DSGE estimation and having troubles replicating their steps in the derivation of demand for final goods firms (Ungated model is here, page 3: <a href="http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=1361&amp;context=soe_research" rel="nofollow">http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=1361&amp;context=soe_research</a>). Their final goods producers maximize profit function:</p>

<p>$\Pi_t = P_t (\int_0^1 Y_t(j)^{1-\nu} dj)^\frac{1}{1-\nu} - \int_0^1 P_t (j)  Y_t(j) dj$</p>

<p>and get the demand:</p>

<p>$Y_t(j) = (\frac{P_t(j)}{P_t})^{-\frac{1}{\nu}} Y_t$</p>

<p>My trouble is that I cannot understand derivation steps: how FOC looks like and why? Help is much appreciated!</p>

<macroeconomics><mathematical-economics>
Derivation of demand for intermediate goods in DSGE model
Daniel Shestakov
# Derivation of demand for intermediate goods in DSGE model

(Asked by: username_0 on 2016-08-16)

I'm self-studying Herbst-Schorfheide book on DSGE estimation and having troubles replicating their steps in the derivation of demand for final goods firms (Ungated model is here, page 3: <http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=1361&context=soe_research>). Their final goods producers maximize profit function:

$\Pi\_t = P\_t (\int\_0^1 Y\_t(j)^{1-\nu} dj)^\frac{1}{1-\nu} - \int\_0^1 P\_t (j) Y\_t(j) dj$

and get the demand:

$Y\_t(j) = (\frac{P\_t(j)}{P\_t})^{-\frac{1}{\nu}} Y\_t$

My trouble is that I cannot understand derivation steps: how FOC looks like and why? Help is much appreciated!

**Tags:** macroeconomics, mathematical-economics


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## 2 answers

### Answer by username_1 on 2016-09-20. Score: 1

To solve the problem you also need the equation:

$Y\_t=\left(\int\_0^1 Y\_t(j)^{1-\nu} \ dj \right)^{\frac{1}{1-\nu}}$

Now, the FOC is the same for every $Y\_t(j)$, so we only have to differentiate once. Applying the chain rule we get:

$\frac{\partial \Pi\_t}{\partial Y\_t(j)}=P\_t \left(\int\_0^1 Y\_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}} Y\_t(j)^{-\nu}-P\_t(j)=0$

At this point, the trick of the trade is that

$\left(\int\_0^1 Y\_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}}$ which is a part of the FOC, equals $Y\_t^{\nu}$ (check the first equation)

Setting
$\left(\int\_0^1 Y\_t(j)^{1-\nu} \ dj \right)^{\frac{\nu}{1-\nu}}= Y\_t^{\nu}$ in the FOC, and solving for $Y\_t(j)$ will give the desired result.


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### Answer by username_2 on 2016-08-18. Score: 1

So I've seen this sort of thing where you basically differentiate with respect to each Y\_j so the integral drops out (seen this in moral hazard chapter from
MWG and Romer's endogenous growth model).

The foc becomes:

$P\_t(Y\_t(j)^{(1-v)})^{v/(1-v)}Y^{-v}-P\_t(j)=0$

This gets the result.


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