C172 flight Instructor reported a NMAC in the pattern at a non-towered airport.

Narrative

The problem arose from what I believe was a lack of communication and pilot negligence. My student and I were doing traffic pattern laps at ZZZ where each leg we called out our position and confirmed with other traffic that was in the pattern or entering the pattern. During this one lap as we had turned left base for Runway XX there I noticed an aircraft on final that was at our 2 o'clock and no more than 1 to 200 feet below us. At that time I took over the controls and executed a climbing left turn back into pattern altitude and said so over the CTAF. The aircraft inbound on final then radio called 'the traffic off our left wing is clear of us' and then continued inbound on final. This aircraft did not make a radio call prior to this that neither my student nor I missed or heard; this was his only transmission so far. After we executed our climbing left turn and were established on final we asked the aircraft if they had ADS-B on board to which they replied they did not. The aircraft was not making any calls and could not be located on the Sentry of G1000 avionics onboard because the aircraft did not have ADS-B installed. The easiest solution to this would have been proper and clear communication; and announcing your position especially in an area that is not controlled. Many accidents happen like this so it is important to communicate especially when you are entering the pattern and critical phases like on a final.