There's another way to think about what I just wrote in the last post. General equilibrium is a lot of objective functions and a lot of budget constraints (including those relevant for the labor market) that have been maximized, so there are a lot of marginal benefits equated to each other and to the vector of prices in the economy.
That's general equilibrium.
That's Say's vision or Malthus's vision plus monetary accommodation to get back to Say's vision.
Full employment is an extra constraint.
Full employment isn't the result of an optimization problem, because unemployment and labor surplus are not the same things. We profess to care about unemployment as the CPS measures it: wanting a job, not having a job, being ready to start a job, and looking for a job. I trust that when people confess to care about this they actually do care about this (I actually do care about this, after all). But this is not necessarily a labor surplus.
So think of a full employment condition as an extra constraint on general equilibrium. Something like:
L = aN, where "a" is pretty stable, between zero and one. We might be tempted to think it is a little lower than the labor force participation rate, but if discouraged workers are prominent it may be higher than the labor force participation rate.
If you care about full employment, you need to add this constraint into general equilibrium - and that creates a problem. We have two options, assuming serendipity is not on our side:
1. Throw something else out of equilibrium to meet this constraint, or
2. Shift some of the other objective functions and budget constraints in order to meet this constraint and be in equilibrium