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.
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