Positive Assortative Matching

Suppose we are given partial orders $\succ_l$ on workers and $\succ_f$ on firms. For instance, $l'\succ_ll''$ might mean that worker $l'$ is more skilled than is worker $l''$, and $f'\succ_ff''$ might mean that a given worker is more productive in firm $f'$ than in firm $f''$.

Theorem. Suppose that for all $l',l'',f'$, and $f''$, $l'\succ_l l''$ and $f'\succ_ff''$ implies that $v_{l'f'}-v_{l''f'}>v_{l'f''}-v_{l''f''}$. Then it cannot be the case that $l'\lra f''$ and $l''\lra f'$.

• The idea of matching bigger with bigger is called positive assortative matching. It came to prominence first in Becker (1973) on marriage.
• The property that the $v$ differences in $l$ are increasing in $f$ is called increasing differences

Proof. If $v_{l'f'}-v_{l''f'}>v_{l'f''}-v_{l''f''}$, then $v_{l'f'}+v_{l''f''}>v_{l'f''}+v_{l''f'}$. So matching $l'$ to $f'$ and $l''$ to $f''$ would increase surplus. ∎