Some labor market puzzles

I had opined that I thought that macro economic models need to address the labor market rather than the product market and I'm wondering if these puzzles are really product market related or just labor market anomalies.

Keep in mind, we have had a recovery in output, but not in employment. That means a smaller number of laborers are working, but we are producing as much as before. As a simple first cut, how should we measure the marginal product of those now laid-off workers? I would start with the number zero. If a restored level of output wouldn't count as evidence for the zero marginal product hypothesis, what would? If I ran a business, fired ten people, and output didn't go down, might I start by asking whether those people produced anything useful?

It is true that the ceteris are not paribus, But the observed changes if anything favor the hypothesis of zero marginal product. There has been no major technological breakthrough in the meantime. If anything, there has been bad monetary policy and a dose of regulatory uncertainty. And yet again we can produce just as much without those workers. Think of "labor hoarding" yet without...the hoarding.

You might cite oligopoly models and argue that the workers can produce something, but firms won't hire them because they don't want to expand output, due to lack of demand. That doesn't seem to explain that output has recovered and that profits are high. And since there is plenty of corporate cash, it is hard to claim that liquidity constraints are preventing the reemployment of those workers.

... In general, which hypotheses predict lots more short-term unemployment among the less educated, but among the long-term unemployed, a disproportionately high degree of older, more educated people? This stylized fact seems to point toward search and recalculation ideas, with some zero marginal products tossed in. Do aggregate demand theories yield that same data-matching prediction? I don't see it, at least not without being paired with a theory of concomitant real shocks.

My question here is what is "real" about shocks? Are we back to exogenous technological shocks again? I hope not.