We study the time constant $C(ν)$ of last passage percolation on the complete directed acyclic graph on the set of non-negative integers, where edges have i.i.d. weights with distribution $ν$ with support included in $\{-\infty\}\cup\mathbb{R}$. We show that $ν\mapsto C(ν)$ is strictly increasing in $ν$. We also prove that $C(ν)$ is continuous in $ν$ for a large set of measures $ν$. Furthermore, when $ν$ is purely atomic, we show that $C(ν)$ is analytic with respect to the weights of the atoms. In the special case of two positive atoms, it is an explicit rational function of these weights.