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In this study, microscopic buckling of elastic square honeycombs subject to in-plane compression is analyzed using a two-scale theory of the up-dated Lagrangian type. The theory allows us to analyze microscopic bifurcation and post-bifurcation behavior of periodic cellular solids. Cell aggregates are taken to be periodic units so that we can discuss the dependence of buckling stress on periodic length. Then, it is shown that microscopic buckling occurs at a lower compressive load as periodic length increases, and that long-wave buckling occurs just after the onset of macroscopic instability if the periodic length is sufficiently long. It is further shown that the macroscopic instability is of the shear type, leading to a simple formula to evaluate the lowest in-plane buckling stress of elastic square honeycombs. [ABSTRACT FROM AUTHOR]