What is the purpose of an emitter resistor?
Answer: Stabilizing the DC bias point against uncertainties, tolerances and temperature changes.
What is the consequence? Reduction in gain (negative feedback effect). However, also the (lower) gain value is stabilized against uncertain transistor parameters and the input-output relation is linearized.
In many cases, we want to achieve a trade-off between stabilization and gain reduction. For this purpose, we realize a good (strong) DC feedback (1kOhm) and a somewhat reduced signal feedback because it is only the 180 Ohm resistor that provides signal feedback (820 Ohm are shorted for signals above the corresponding cutoff frequency using a large emitter capacitor).
Edit/Update: Gain formulas:
Rc=10k; RE1=180=hm; RE2=820Ohm; RE=RE1+RE2
(a) DC and AC (far below the cutoff freqency fc):
G=gmRc/(1+gmRE) with transconductance gm=IC/Vt (Vt=26mv).
(b) For AC (far above fc):
G=gmRc/(1+gmRE1)>>>>>>for RE1>>1/gm=26 Ohm we have app. G~Rc/RE1.
(c) Cutoff frequency: fc=1/(6.28*T) with T=CE*[RE2||(RE1+1/gm)]