what is the interpretation of $\beta_{ij}$ in the translog production function?
You may interpret $\alpha_i$ as the output elasticity regarding input factor $x_i$, keeping other factors $x_j, j\neq i$ constant. $\beta_{ij}$ is an indicator for complementarities in production, i.e. how much does the addition of one percent of both factors $x_i$ and $x_j$ add to $y$. Since there is the factor 0.5, and since $\alpha_i$ and $\alpha_j$ are relevant too, the interpretation is not that straightforward.