Consumer Theory question
You could use the fact that $\sum_{i=1}^{n}e_i=y$ (adding-up property) to see which parameter values are compatible with the above specification and this identity.
Before proceeding, you should replace on the RHS $\sum_{i=1}^{n}\beta_{ij}p_i$ by $\sum_{j=1}^{n}\beta_{ij}p_j$.