In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called $\tau_{\epsilon,\phi}$ algebra, a generalization of the more known $\tau$ algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator $T_{n,\epsilon,\phi}$ of the $\tau_{\epsilon,\phi}$ algebra. In particular, we derive the asymptotics for the outliers of $T_{n,\epsilon,\phi}$ and the associated eigenvectors; we obtain equations for the eigenvalues of $T_{n,\epsilon,\phi}$, which provide also the eigenvectors of $T_{n,\epsilon,\phi}$; and we compute the full eigendecomposition of $T_{n,\epsilon,\phi}$ in the specific case $\epsilon\phi=1$. We also present applications of our results in the context of queuing models, random walks, and diffusion processes, with a special attention to their implications in the study of wealth/income inequality and portfolio dynamics.