# When is expenditure function non-decreasing?

I have to find parameters m and d for which expenditure function is non-decreasing and homogeneous of degree 1. My expenditure function is:

I think that I should find ∂e/∂p which has to be >= 0 but I have a problem with this because expenditure function is for n-good. Can someone help me?

• Your expenditure function is a bit unclear. What is $z$? Also you can use latex to write the equation for clarity. Dec 5, 2020 at 15:03
• Z is a parameter. I changed names of this parameters so as not to cause confusion.
– Azsb
Dec 5, 2020 at 15:30
• There are two ambiguities here: (i) this looks as an exercise... which we are not supposed to solve. (ii) it seems that you use the \textbf{same} notation $e$ for denoting two \textbf{different} things. This may prevent us to answer or to give hints. Dec 6, 2020 at 10:13
• Second "e" in equation is exp.
– Azsb
Dec 6, 2020 at 10:19

One possibility is to reparameterize your expenditure as follow: $$e(\mathbf{p},u) = E(f(\mathbf{p}),g(\mathbf{p})u),$$ with $$f,g:\mathbb{R}^n\rightarrow \mathbb{R}$$. Then use the chain rule to find $$\partial{e}/\partial{p_j}$$, $$\partial{e}/\partial{u}$$ and to prove that $$e$$ is homogeneous of degree one in $$\mathbf{p}$$ iff ...