  We consider the thermodynamic consistency of the charge response function in
the (extended) Hubbard model. In DMFT, thermodynamic consistency is preserved.
We prove that the static, homogeneous DMFT susceptibility is consistent as long
as vertex corrections obtained from the two-particle impurity correlation
function are included. In presence of a nonlocal interaction, the problem may
be treated within extended DMFT (EDMFT), or its diagrammatic extension, the
dual boson approach. We show that here, maintaining thermodynamic consistency
requires knowledge of three- and four-particle impurity correlation functions,
which are typically neglected. Nevertheless, the dual boson approximation to
the response is remarkably close to consistency. This holds even when
two-particle vertex corrections are neglected. EDMFT is consistent only in the
strongly correlated regime and near half-filling, where the physics is
predominantly local.
