Expenditure function
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.