<p>The Lagrangian is the following:
<span class="math-container">$$
L = \left(\int_0^1 C_t(i)^{1 - \frac{1}{\varepsilon}}di \right)^{\frac{\varepsilon}{\varepsilon-1}} - \lambda\left(\int_0^1 P_t(i) C_t(i) di - Z_t\right)
$$</span></p>
<p>Let <span class="math-container">$\left(\int_0^1 C_t(i)^{1 - \frac{1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}} = C_t$</span>.</p>
<p>The first order conditions are given by:
<span class="math-container">$$
\frac{\varepsilon}{\varepsilon-1}\left(\int_0^1 C_t(j)^{1 - \frac{1}{\varepsilon}} dj \right)^{\frac{1}{\varepsilon-1}}\left(1 - \frac{1}{\varepsilon}\right)C_t(i)^{-\frac{1}{\varepsilon}} - \lambda P_t(i) = 0
$$</span></p>
<p>This can be rewritten as:
<span class="math-container">$$
\begin{align*}
&amp;C_t^{\frac{1}{\varepsilon}} C_t(i)^{-\frac{1}{\varepsilon}}  = \lambda P_t(i),\\
\leftrightarrow &amp;\left(\frac{C_t(i)}{C_t}\right)^{-\frac{1}{\varepsilon}} = \lambda P_t(i).
\end{align*}
$$</span></p>

CC BY-SA 4.0
2024-02-02T12:29:59.777Z
Thank's @tdm ! So if I understand it well, the integrals doesn't have any impact on the derivation process ?
justaneconomist
CC BY-SA 4.0
2024-02-02T14:22:35.777Z
Yes, I always struggled myself with this 'leap of faith' derivation... but that's how they do it :-)
tdm
CC BY-SA 4.0
2024-01-31T14:04:30.450Z
2024-01-31T14:04:30.450Z
tdm
<p>I am currently studying econ models, and trying to calculate the FOC for different kind of models. I use J.Gali  &quot;Monetary Policy, Inflation and the Business Cycle - An Introduction to the New Keynesian Framework&quot;.</p>
<p>In the second chapter, the classical monetary model households have the following objective function :</p>
<p><span class="math-container">$ E_0\sum^\infty_{t=0} \beta^t U(C_t, N_t)$</span></p>
<p>under the budget constraints</p>
<p><span class="math-container">$P_t C_t + Q_t B_t \le B_{t-1}+W_tN_t-T_t$</span></p>
<p>I completely understand how to find the FOC under this model.</p>
<p>However, in the next chapter, households have the same utility function <span class="math-container">$ E_0\sum^\infty_{t=0} \beta^t U(C_t, N_t)$</span>. However, here <span class="math-container">$ C_t \equiv(\int^1_0 C_t(i)^{1-\frac{1}{\varepsilon}}di)^{\frac{\varepsilon}{\varepsilon-1}} $</span>.
Households try to maximize their utility under <span class="math-container">$P_tC_t+Q_tB_t\le B_{t-1}+W_tN_t+T_t$</span></p>
<p>Where <span class="math-container">$P_tC_t=\int^1_0P_t(i)C_t(i)di$</span></p>
<p>If I understand well, in this kind of model we have a continuum of goods <span class="math-container">$i$</span> and households face an optimal allocation of consumption expenditures problem. Before maximizing their utility depending on both <span class="math-container">$C_t$</span> and <span class="math-container">$N_t$</span>, they have to maximize <span class="math-container">$C_t$</span> for any given expenditure level (<span class="math-container">$\int^1_0P_t(i)C_t(i)\equiv Z_t$</span> where <span class="math-container">$Z_t$</span> is the expenditure level), so the maximization problem yields</p>
<p>L= [<span class="math-container">$ \int^1_0 C_t(i)^{1-\frac{1}{\varepsilon}}di]^{\frac{\varepsilon}{\varepsilon-1}} - \lambda(\int^1_0P_t(i)C_t(i)di- Z_t) $</span></p>
<p>Intuitively, I'd replace <span class="math-container">$\int^1_0 C_t(i)$</span> by <span class="math-container">$C_t$</span> and do <span class="math-container">$\frac{dL}{dC_t}=0$</span> which would give me something like <span class="math-container">$\lambda_tP_t =  \frac{\varepsilon}{\varepsilon -1}C_t^{\frac{\varepsilon}{\varepsilon -1}-1}$</span></p>
<p>But here you see I don't really take in account the integrals, and the answer given in the book is :
<span class="math-container">$C_t(i)^{-{1/\varepsilon}} C_t^{1/\varepsilon}=\lambda_t P_t(i)$</span></p>
<p>I don't know how to find this and I'm guessing it's linked to the integrals derivations. What is the process behind this ?</p>

CC BY-SA 4.0
2023-12-24T13:57:03.983Z
Did you look at the chapter appendix?
Michael Greinecker
CC BY-SA 4.0
2023-12-24T13:58:00.747Z
Yep, and actually what I've written here is in appendix for some part. It just gives the results, not the demonstrations.
justaneconomist
<inequality><lagrangian><model>
Dealing with integrals in econ models
justaneconomist
# Dealing with integrals in econ models

(Asked by: username_0 on 2023-12-24)

I am currently studying econ models, and trying to calculate the FOC for different kind of models. I use J.Gali "Monetary Policy, Inflation and the Business Cycle - An Introduction to the New Keynesian Framework".

In the second chapter, the classical monetary model households have the following objective function :

$ E\_0\sum^\infty\_{t=0} \beta^t U(C\_t, N\_t)$

under the budget constraints

$P\_t C\_t + Q\_t B\_t \le B\_{t-1}+W\_tN\_t-T\_t$

I completely understand how to find the FOC under this model.

However, in the next chapter, households have the same utility function $ E\_0\sum^\infty\_{t=0} \beta^t U(C\_t, N\_t)$. However, here $ C\_t \equiv(\int^1\_0 C\_t(i)^{1-\frac{1}{\varepsilon}}di)^{\frac{\varepsilon}{\varepsilon-1}} $.
Households try to maximize their utility under $P\_tC\_t+Q\_tB\_t\le B\_{t-1}+W\_tN\_t+T\_t$

Where $P\_tC\_t=\int^1\_0P\_t(i)C\_t(i)di$

If I understand well, in this kind of model we have a continuum of goods $i$ and households face an optimal allocation of consumption expenditures problem. Before maximizing their utility depending on both $C\_t$ and $N\_t$, they have to maximize $C\_t$ for any given expenditure level ($\int^1\_0P\_t(i)C\_t(i)\equiv Z\_t$ where $Z\_t$ is the expenditure level), so the maximization problem yields

L= [$ \int^1\_0 C\_t(i)^{1-\frac{1}{\varepsilon}}di]^{\frac{\varepsilon}{\varepsilon-1}} - \lambda(\int^1\_0P\_t(i)C\_t(i)di- Z\_t) $

Intuitively, I'd replace $\int^1\_0 C\_t(i)$ by $C\_t$ and do $\frac{dL}{dC\_t}=0$ which would give me something like $\lambda\_tP\_t = \frac{\varepsilon}{\varepsilon -1}C\_t^{\frac{\varepsilon}{\varepsilon -1}-1}$

But here you see I don't really take in account the integrals, and the answer given in the book is :
$C\_t(i)^{-{1/\varepsilon}} C\_t^{1/\varepsilon}=\lambda\_t P\_t(i)$

I don't know how to find this and I'm guessing it's linked to the integrals derivations. What is the process behind this ?

**Tags:** inequality, lagrangian, model

> **Comments:**
> **username_1**: Did you look at the chapter appendix?  
> **username_0**: Yep, and actually what I've written here is in appendix for some part. It just gives the results, not the demonstrations.  

---

## 1 answers

### Answer by username_2 on 2024-01-31. Score: 4 | Accepted answer

The Lagrangian is the following:
$$
L = \left(\int\_0^1 C\_t(i)^{1 - \frac{1}{\varepsilon}}di \right)^{\frac{\varepsilon}{\varepsilon-1}} - \lambda\left(\int\_0^1 P\_t(i) C\_t(i) di - Z\_t\right)
$$

Let $\left(\int\_0^1 C\_t(i)^{1 - \frac{1}{\varepsilon}}\right)^{\frac{\varepsilon}{\varepsilon-1}} = C\_t$.

The first order conditions are given by:
$$
\frac{\varepsilon}{\varepsilon-1}\left(\int\_0^1 C\_t(j)^{1 - \frac{1}{\varepsilon}} dj \right)^{\frac{1}{\varepsilon-1}}\left(1 - \frac{1}{\varepsilon}\right)C\_t(i)^{-\frac{1}{\varepsilon}} - \lambda P\_t(i) = 0
$$

This can be rewritten as:
$$
\begin{align\*}
&C\_t^{\frac{1}{\varepsilon}} C\_t(i)^{-\frac{1}{\varepsilon}} = \lambda P\_t(i),\\
\leftrightarrow &\left(\frac{C\_t(i)}{C\_t}\right)^{-\frac{1}{\varepsilon}} = \lambda P\_t(i).
\end{align\*}
$$

> **Comments:**
> **username_0**: Thank's @username_2 ! So if I understand it well, the integrals doesn't have any impact on the derivation process ?  
> **username_2**: Yes, I always struggled myself with this 'leap of faith' derivation... but that's how they do it :-)  

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