According to al-Jaziri, if the Imam does not offer a sacrifice, then the sacrifice may be made by the common people after an interval of time has passed in which it would have been possible for the Imam to have made the sacrifice. This would be the case if the Imam is not able to afford it.
المالكية قالوا: يبتدئ وقت الأضحية لغير الإمام في اليوم الأول بعد تمام ذبح الإمام. ويبتدئ وقتها للإمام بعد الفراغ من خطبته بعد صلاة العيد، أو مضي زمن قدر ذبح الإمام أضحيته إن لم يذبح الإمام
According to the Malikis, the time for slaughtering one's udhiyah for people
other than the imam begins on the Day of Sacrifice after the imam has slaughtered
his. The time for doing so for the imam himself begins after he has completed his
sermon following the holiday prayer, or after sufficient time for the imam to have
slaughtered his udhiyah has passed if he does not, in fact, slaughter an udhiyah.
— الفقه على المذاهب الأربعة - [English translation]
As far as I can tell this condition regarding the starting time for sacrifice exists in the Maliki madhab. The others generally hold the view that the time begins after Eid prayer, see for example the view of the other madhabs in the reference above and in الفقه الإسلامي وأدلته and الموسوعة الفقهية and islamweb.