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?

  • $\begingroup$ Your expenditure function is a bit unclear. What is $z$? Also you can use latex to write the equation for clarity. $\endgroup$
    – Dayne
    Dec 5, 2020 at 15:03
  • $\begingroup$ Z is a parameter. I changed names of this parameters so as not to cause confusion. $\endgroup$
    – Azsb
    Dec 5, 2020 at 15:30
  • $\begingroup$ 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. $\endgroup$
    – Bertrand
    Dec 6, 2020 at 10:13
  • $\begingroup$ Second "e" in equation is exp. $\endgroup$
    – Azsb
    Dec 6, 2020 at 10:19

1 Answer 1


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 ...


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