# Lagrange with integral

1. Apr 16, 2013

### Charlotte87

1. The problem statement, all variables and given/known data
Maximize $C_{t}$ for any given expenditure level

$\int_{0}^{1}P_{t}(i)C_{t}(i)di\equiv Z_{t}$

3. The attempt at a solution

The Lagrangian is given by:
$L = \left(\int_{0}^{1}C_{t}(i)^{1-(1/\varepsilon)}di\right)^{\varepsilon/(\varepsilon-1)} - \lambda \left(\int_{0}^{1}P_{t}(i)C_{t}(i)di - Z_{t}\right)$

I know that the first order condition is

$C_{t}(i)^{-1/\varepsilon}C_{t}^{1/\varepsilon} = \lambda P_{t}(i)$ for all $i \in (0,1)$

But I do not understand how they get to this answer. Can anyone help me?

Last edited by a moderator: Apr 16, 2013
2. Apr 16, 2013

### HallsofIvy

Staff Emeritus
3. Apr 16, 2013

### Charlotte87

No, I have to admitt I've never heard of it...