Good evening, I have to solve the following minimization problem: $$\min_{w(p)}: \int w(p)f(p|e)dp$$
s.t.
$$\int v(w(p))f(p|e)dp - g(e) ≥ u_r$$
where of course this last inequality is my constraint. I think it is irrelevant for my problem to specify the meaning of the functions, but I know also that $v’(w) > 0 $ and $v’’(w) ≤ 0$.
I proceed in this way, first I create the Lagrangian: $$L=\int [w(p)f(p|e)dp]+λ[v(w(p))f(p|e)dp - g(e)-u_r]$$
At this point I have to take the derivative with respect to $w(p)$ and pose it equal to zero
$$ \frac{\partial L} {\partial (w(p))}=0$$
at this point I don't know how to procede; my problem is how treat those integrals. The expected solution (first order condition) is:
$$-f(p|e)+\lambda v'(w(p))f(p|e)=0$$
Thank you very much. PS: I know also that p takes value in $[p_d,p_u]$
Edit: the first order condition for $w(p)$ is derived taking the derivative with respect to w at each level of $p$ separately. To see this point consider a discrete version of the model in which there is a finite number of possible p levels $(p_1,...,p_N)$ and associated w levels $(w_1,...,w_N)$. The first order condition is analogous to the condition one gets in the discrete model by examining the first order conditions for each $w_n=1,...,N$. TO be rigorous we shoud add that when we have a continuum of possible level of p, an optimal scheme need only to satisfy condition at a set of profit levels thati is full of measure.