The effective embedding potential introduced by Wesolowski and Warshel [J. Phys. Chem., 97 (1993) 8050] depends on two electron densities: that of the environment (n B ) and that of the investigated embedded subsystem (n A ). In this work, we analyze this potential for pairs n A and n B, for which it can be obtained analytically. The obtained potentials are used to illustrate the challenges in taking into account the Pauli exclusion principle.