Preamble, set-up: In economics, a well known optimization problem writes as follows: $$ \max_{\{x\}_{j=1}^n,y}U(x_1,y)+\sum_{i=2}^{n}\lambda_i\cdot U(x_i,y)+\mu\cdot(y-\sum_{j=1}^{n}(w_j-x_j)) $$
where $\lambda_i$ and $\mu$ are (binding) lagrangian multipliers such that $\mu>0$, and $\lambda_i>0$ for all $i$, and $U$ is a differentiable and concave function.
An easy way to solve this problem taking first order conditions: $$ \{x_j\}:\lambda_j\cdot\frac{\partial U(x_j,y)}{\partial x_j}+\mu=0, \quad \text{for $j=1,\dots,n$, and $\lambda_1=1$} $$ $$ \{y\}:\sum_{j=1}^n\lambda_j\cdot\frac{\partial U(x_j,y)}{\partial y}+\mu=0, \quad \text{for $j=1,\dots,n$, and $\lambda_1=1$} $$
Notice that provided $\mu>0$ we can divide the last equation by $\mu$ and the use the $n$ firs order conditions for $x_j$ in order to get a compact necessary condition:
$$ \sum_{j=1}^n\frac{{\partial U(x_j,y)}/{\partial y}}{{\partial U(x_j,y)}/{\partial x_j}}=1 $$
My interest: I would like to expand this result to a setting were there is an infinite number of consumers. Lets say that consumers are indexed by $j\in[0,1]$ according to an atomless CDF $F(j)$. Essentialy, I would like to retrieve an equivalent necesary optimality condition like:
$$ \int_0^1\frac{{\partial U(x_j,y)}/{\partial y}}{{\partial U(x_j,y)}/{\partial x_j}}dF(j)=1 $$
The optimization problem would look like:
$$ \max_{\{x\}_{j\in[0,1]},y}\int_{0}^1\lambda_j\cdot U(x_j,y)dF(j)+\mu\cdot\left(y-\int_{0}^1(w_j-x_j)dF(j)\right) $$
However if I want to retrieve the desired expression it seems like I need that:
$$ \frac{\int_0^1(w_i-x_i)dF(i)}{dx_i}=-1 $$
and I do not know under which conditions I can get this. Any suggestions? Thanks!