  We have studied the chemical potential shift in the high-temperature
superconductor Bi$_2$Sr$_2$Ca$_{1-x}${\it R}$_{x}$Cu$_2$O$_{8+y}$ ({\it R} =
Pr, Er), where the hole concentration is varied from 0.025 to 0.17 per Cu, by
precise measurements of core-level photoemission spectra. The result shows that
the shift becomes slow in the underdoped region as in the case of
La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO) but the effect is much weaker than in LSCO.
The observed shift in the present system can be relatively well explained by
numerical results on the doped two-dimensional Hubbard model, and suggests that
the change of the electronic structure induced by hole doping is less
influenced by stripe fluctuations than in LSCO.
