I am trying to solve what looks like a simple system of equations:
$$x_j = A_j\left(\sum_{i=1}^n B_{ij} x_i\right)^\alpha $$
for all $j\in\{1,\dots,n\}$, where $n$ is a positive integer, $0<\alpha<1$ and all the $A_j\geq 0$ and $B_{ij}\geq 0$ are known constants. I can also impose $B_{ii}=0$ for all $i$ if it helps. The system has a trivial solution at $x=0$. Because of the concavity on the right-hand side, I believe that there is also a unique non-trivial solution $x^*$. Since the system is so simple, I'm looking for a closed-form solution for $x^*$ but I haven't been able to find it. Any help would be greatly appreciated! Thanks!