I am triying to solve the following functional maximization problem:
$\max_{x(\theta)} \int_0^{x(\theta)]+c} [\theta-x(\theta)]x(\theta)d\theta + \int_{x(\theta)]+c}^1 \big[\frac{\theta-x(\theta)+c}{2} \big]x(\theta)d\theta$
subject to:
$\int_0^1 x(\theta) d\theta=0$
I think that I cannot use pointwise maximzation, since the boundaries of the integral are not a function of $x(\theta)$. How do I solve this problem then?
One idea (that is probably wrong) is to realize that the problem above is equal to
$\max_{x(\theta)} \int_0^{1} \min \big\{\theta-x(\theta), \frac{\theta-x(\theta)+c}{2}\big\} x(\theta)d\theta$
subject to:
$\int_0^1 x(\theta) d\theta=0$
Then, I can look for the optimal solution by solving the following two lagrangians:
$\max_{x(\theta)} \int_0^{1} \big\{\theta-x(\theta)\big\} x(\theta)d\theta + \lambda \int_0^1 x(\theta) d\theta$
and:
$\max_{x(\theta)} \int_0^{1} \frac{\theta-x(\theta)+c}{2} x(\theta)d\theta + \lambda \int_0^1 x(\theta) d\theta$
It turns put that the solution to the two problems is the same, and it is given by $x(\theta) = \frac{\theta}{2}-\frac{1}{4}$
Does this make any sense? What do you think?