Could anyone please give any idea on how the following system of equations has to be solved with respect to vector $\hat W$ (which, essentially, contains all the unknowns in the system).
\begin{cases} \hat L_{i,j} = (\sigma - 1)\sum_{k=1}^9\nu_{k,j}\hat p_{k,j} \space - \space \sigma\hat p_{i,j},\\ \hat p_{k,j}=\tau_{k,j} + \hat w_{k},\\ \sum_{j=1}^9s_{i,j}\hat L_{i,j} = \beta\hat w_i \end{cases}
The model describes change in employment $\hat L_{i,j}$ in region i because of a change in supplies to region j.
$\sigma, \beta $ are exogenous constants. $\nu$ is a matrix of constants ($\nu_{k,j}$ is a share of region's j income spent on goods from region k), $s$ is a matrix of constants ($s_{i,j}$ is a share of region's i production directed to region j), $\tau$ is a matrix of constants as well ($\tau_{k,j}$ represents costs related to trade between region k and j). $p$'s are prices of goods and $w$'s are wages in corresponding regions. There are 9 regions in total.
I can't figure out the way to find a general solution to the system (preferably, in matrix form, which I can't derive for this system either). Writing them all down doesn't really seem rational since there're like 81+81+9 equations.
If there's no general solution, how do I aproach it then to find vector $\hat W$? Any ideas? Thanks in advance!
$$ \sum_{j=1}^9s_{i,j}\left((\sigma - 1)\sum_{k=1}^9\nu_{k,j}\left(\tau_{k,j} + \hat w_{k}\right) \space - \space \sigma( \tau_{i,j} + \hat w_{i})\right) = \beta\hat w_i $$
$$ (\sigma - 1)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\tau_{k,j}\right)+(\sigma - 1)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\hat w_{k}\right)-\sigma\sum_{j=1}^9s_{i,j}\tau_{i,j}-\sigma\sum_{j=1}^9s_{i,j} \hat w_{i} = \beta\hat w_i $$
$$ \beta\hat w_i+\sigma\sum_{j=1}^9s_{i,j} \hat w_{i}+(\sigma - 1)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\hat w_{k}\right) = (1-\sigma)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\tau_{k,j}\right)-\sigma\sum_{j=1}^9s_{i,j}\tau_{i,j} $$
$$\left(\beta+\sigma\sum_{j=1}^9s_{i,j}\right)\hat w_i+(\sigma-1)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\hat w_{k}\right) = (1-\sigma)\sum_{j=1}^9s_{i,j}\left(\sum_{k=1}^9\nu_{k,j}\tau_{k,j}\right)-\sigma\sum_{j=1}^9s_{i,j}\tau_{i,j}$$
Now define $S$, $V$, $T$, $\hat{\mathbf w}$ as matrices/vectors where $[S]_{i,j}=s_{i,j}$, $[V]_{k,j}=\nu_{k,j}$, $[T]_{i,j}=\tau_{i,j}$, $[\hat{\mathbf w}]_{i}=\hat w_{i}$, as well as $\mathbf 1=[1,1,1,\ldots,1]^t$ (a column vector of 9 ones) and I think we have (you should check the details)
$$\left(\beta+\sigma \mathbf 1^tS^t+(\sigma-1)SV^t\right)\hat{\mathbf w} = (1-\sigma)S\mathbf 1^t(V\circ T)-\sigma(S\circ T)\mathbf 1$$
where $A\circ B$ is the Hadamard product (element-wise multiplication). This is a linear system but I don't think you can expect to find a general solution, at least it's a simple expression to give to a numerical solver.