To investigate the relation between the macroscopic and microscopic instability predictions for certain microstructured media, we study the stability of an axially stretched fiber-reinforced composite under plane strain conditions. The microstructural instability is attributed to a bifurcation buckling of the fibers while the corresponding macroscopic one occurs at the loss of ellipticity in the homogenized incremental equilibrium equations of the material. The effects of geometry and material properties on those phenomena are analyzed. The macroscopic approach consistently and sometimes considerably overestimates the stable region of the composite. The attractive feature in this work is that all the investigations can be done by using analytical solutions instead of the numerical ones employed in similar investigations so far.