Why does adding an emitter resistor to a common emitter amp reduce its gain?
Gain is the output voltage change divided by the input voltage change.

In the small signal hybrid pi model the collector current changes proportional to the change in base voltage (relative to the emitter) so there is a voltage change at the collector proportional to the load resistance (parallel with ro).



If you add an emitter resistor, the base-emitter voltage does not change as much so the current change is less and therefore the gain is reduced. That is because the emitter current change causes the emitter voltage to change in the same direction as the change in base voltage, partially counteracting the effect.