Recently, a new model describing the cell dynamics in hematopoiesis was proposed. It can be described as an age-structured delay differential-difference model. Under some conditions on the biological parameters, it admits two equilibrium points. The first one is the 0-equilibrium and the second one, which does not always exist, is a strictly positive point. We propose a Lyapunov functional construction in order to investigate the stability properties of both equilibria. For the 0 equilibrium, we establish the global exponential stability when the positive equilibrium does not exist. For the positive equilibrium, we establish its local exponential stability, estimate the decay rate of solutions and provide an open subset of its basin of attraction.