A Criterion for Contiguity of Quasiconcave Functions

Quasiconcave functions $\rho _0$ and $\rho _1$ belong to the same scale if there exist quasiconcave functions $\psi _0$ and $\psi _1$ and numbers $0<\theta _0<1$, $0<\theta _1<1$ such that $\rho _0=\psi _0^{1-\theta _0}\psi _1^{\theta _0}$ and $\rho _1=\psi _0^{1-\theta _1}\psi _1^{\theta _1}$. We establish a criterion for such functions to belong to the same scale up to equivalence. This criterion is obtained in terms of nodes of the corresponding linear-constant step-functions. It turns out that nodes must be equivalent to sequences.