Let $S$ be an $N\times N$ row-stochastic matrix and let $w'$ be the left Perron eigenvector of $S$ (i.e., $w$ is the stationary distribution of the Markov chain represented by $S$). Let $T$ be the matrix whose $in$-th entry is $S_{ni}w_{i}/w_{n}$. $T$ is also a row-stochastic matrix.
For any vector $\left\{ \hat{z}_{i}\right\} _{i=1}^{N}>0$, we look for $\left\{ \hat{w}_{i}\right\}$ that form the solution to the following system of equations:
$ \hat{w}_{i}=\sum_{n}T_{in}\hat{w}_{n}\frac{\left(\hat{w}_{i}/\hat{z}_{i}\right)^{-1}}{\sum_{k}\left(\hat{w}_{k}/\hat{z}_{k}\right)^{-1}S_{nk}}.$
Claim: the vector $(\hat{w}_i)$ is log-linear in $(\hat{z})$. That is, there exists a matrix $M$ such that the vector $\ln\hat{w}\equiv(\ln\hat{w}_i)$ is equal to $M\ln\hat{z}$.
I tried taking log of the RHS and taking a first-order approximation, which by definition creates a linear relationship between $\ln\hat{w}$ and $\ln\hat{z}$. Based on numerical solutions of the nonlinear equations, the first-order approximation seems to coincide with the numerical solution, but I don't know how to prove this.
