  An atomistic effective Hamiltonian is used to compute electrocaloric (EC)
effects in rare-earth substituted BiFeO$_{3}$ multiferroics. A phenomenological
model is then developed to interpret these computations, with this model
indicating that the EC coefficient is the sum of two terms, that involve
electric quantities (polarization, dielectric response), the antiferromagnetic
order parameter, and the coupling between polarization and antiferromagnetic
order. The first one depends on the polarization and dielectric susceptibility,
has the analytical form previously demonstrated for ferroelectrics, and is thus
enhanced at the ferroelectric Curie temperature. The second one explicitly
involves the dielectric response, the magnetic order parameter and a specific
magnetoelectric coupling, and generates a peak of the EC response at the N\'eel
temperature. These atomistic results and phenomenological model may be put in
use to optimize EC coefficients.
