# Stable Outcomes and Matching Models for the Labor Market

I am reading a paper that discusses mathcing of workers to firms in the labor market (see the excerpt below). Here an outcome $$(\mu,\pi,w)$$ consists of a matching function pairing workers to firms, a profit vector for firms, and a wage vector for workers. The text states that the outcome is stable if there is no blocking pair and that this happens if $$w(x) + \pi(y) >= f(x,y) \forall (x,y)$$. My question is: how does this condition ensure a stable outcome?

I understand that this condition requires that the sum of the wage and profit in the existing pair be larger than any pair. And I understand that this means that a firm can always increase wages to keep their preferred worker from deviating to another firm. But that implies that the matching function mu is stable. I don't understand what in the above condition states that the wage and profit vectors in the stable outcome are actually selected to be stable?

Thank you!

You don't want a worker $$x$$ and a firm $$y$$ to be able to give up on who they were previously matched with, match, and be better off. Currently, the worker receives $$w(x)$$ and the firm $$\pi(y)$$. If they would be better off together, they need to generate more surplus than $$w(x)+\pi(y)$$. This requires $$f(x,y)>w(x)+\pi(y)$$. That $$f(x,y)\leq w(x)+\pi(y)$$ for all $$x$$ and $$y$$ means this is not possible, and the matching must be stable.
• I understand your point which implies that the match would be stable. However, why does the paper claim that the outcome $(\mu,\pi,w)$ is stable? Would not a stable outcome require that the firm actually pay a competitive wage? For example, if worker $i$ is matched with a firm and makes a huge surplus, but worker $i$ is paid $w_i = 0$ the match may be stable but the outcome may not be. Yet the paper I quoted claims that the outcome is stable. I have seen one other paper make the same claim so I am not sure whether it is a mistake or not.
• Whether a worker being paid $0$ is compatible with a stable outcome will depend on the details. Maybe you could add all relevant definitions and the actual reference to the paper. Oct 11, 2023 at 18:00