Expenditure function Let $u (x) = \prod_{{i\leq n}}$ $(x_i-m_i)^{a_i}$, where $m_i\geq$0 and $a_i\geq$0, Σ$_{{i\leq n}}a_i=1$ show that the expenditure function $e(p,u)$ is linear in $u$.
Based on the definition of the EF, my first guess would be to find the MRS in terms of $x_i$ and $m_i$ to we get $x_i^*$ and $m_i^*$. After we find these two in terms of each other, we substitute it back to the $u$ function to get now $x_i^*$ and $m_i^*$ in terms of $u$ as well. After this we substitute what we found to the budget constraint and prove that the expenditure function is linear in $u$ somehow.
I don't know if I should follow this logic or is there some other way to approach this problem? Any help is appreciated.
It is not quite true; the expenditure function is affine in utility, not linear. This is Stone-Geary utility, and I use the result that demand is given by
$$x_i^*=m_i+\frac{a_i}{p_i}\bigg(w-\sum_j m_jp_j\bigg).$$
One can interpret $\sum_j m_jp_j$ as &quot;subsistence expenditure&quot; and $w-\sum_j m_jp_j$ as &quot;non-subsistence expenditure.&quot; Plugging this into the utility function, one can show that utility is an increasing linear function of non-subsistence expenditure. The inverse of a linear function is linear again, so we can write non-subsistence expenditure as $l(u)$ with $l$ linear (in the relevant range). This means then that total expenditure is
$$l(p)+\sum_j m_jp_j,$$
which is an affine function as the sum of a  linear function and a constant.