I'm study Economics and I have a math question about the first order condition on an integral. Let's me explain:
First, the Lagrangian for the problem (Optimal Allocation of Consumption Expenditures Dixit-Stiglitz Model) is:
$$\mathcal{L} = \left[\int_0^1 C_t(i)^{1-\frac{1}{\theta}} di\right]^{\theta /(\theta-1)} - \lambda \left(\int_0^1 P_t(i) C_t(i) di \right)$$
The first order condition in the book is something like this:
$$\left[\int_0^1 C_t(i)^{1-\frac{1}{\theta}} di\right]^{1 /(\theta-1)} C_t(i)^{-1/\theta} = \lambda P_t(i)$$
Do you know why? I tried leibniz formula, fundamental theorem of calculus, but nothing works.
Thank you !
I'm reading the question now and I think the FOC could be obtained with derivation rules, even if integrals make the expression apparently difficoult. We have to find the optimal quantity consumed of every good in our model so we need to derive the lagrangian respect to $C_t(i)$ so:
$\displaystyle\frac {d\mathcal{L}}{dC_t(i)}=0$ and it is equal to $\displaystyle \frac{\theta}{\theta-1}\Big[\int_{0}^{1}C_t(i)^{1-\frac{1}{\theta}}di\Big]^{\frac{\theta-\theta+1}{\theta-1}}\cdot \Big(1-\frac{1}{\theta}\Big)C_t(i)^{1-\frac{1}{\theta}-1} -\lambda P_t(i)=0$
I think the reason because the summation is not present in the derivative is that we have to derive respect all $C_t(i)$ and here, we have an infinite quantity of them, because we have an integral, so we write the result for one of them, I think. When we derive the integral I think we can deal with it as we are dealing with a normal function because here we're supposing that the quantity consumed for each variety increases a little bit, so we have to do a integral over a quantity consumed a little bit higher of each variety $C_t(i)$. Leibniz's rule I think it can't be used because is for integrand functions with two variables. Maybe you forgot the share of income for the manufacturing good into the parenthesis multiplied by $\lambda$ from which you've to subtract $\displaystyle \int_{0}^{1} P_t(i)C_t(i) di$. Please someone correct me if I made mistakes because I'm not 100% sure of everything I've written.
What's the name of the book you're using?