Gamma-ratio marginal and profile likelihood identity ($\ell (\psi ;t)=m\log \psi -(m+n)\log (1+\psi t)$)

For independent $X\sim \mathrm{Gamma}(m,\psi \lambda )$ and $Y\sim \mathrm{Gamma}(n,\lambda )$ in shape-rate notation, $T=X/Y$ has density proportional to ${\psi }^m{t}^{m-1}(1+\psi t{)}^{-(m+n)}$. Its distribution is nuisance-free in lambda. The nuisance maximum is $(m+n)/(\psi X+Y)$, so substituting into the full likelihood gives the same interest-parameter log-likelihood up to a parameter-free term. Equality here is special to the common-scale structure, not a general equality of profiling and marginalization.