# 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.

• There is no $m_i^*$; $m_i$ is a fixed parameter. Nov 29 '20 at 23:25
• It makes sense because I wasn't going anywhere with that :/, but I'm still facing problems deriving it. Do you have any clues as to what may be the first step? The book I have isn't of any help either. Thanks in advance) Nov 30 '20 at 11:20
• Having tried it myself, I think it is not true unless we are in the Cobb-Douglas case with $m_i=0$ for all $i$. This is Stone-Geary utility, you can find some background here. Nov 30 '20 at 12:43

$$x_i^*=m_i+\frac{a_i}{p_i}\bigg(w-\sum_j m_jp_j\bigg).$$ One can interpret $$\sum_j m_jp_j$$ as "subsistence expenditure" and $$w-\sum_j m_jp_j$$ as "non-subsistence expenditure." 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.