Explaining negative “cyclic” unemployment with a production function

First of all, here is a related post. Unfortunately, it does not answer my question.

I have heard that there is an understanding of negative unemployment in economic theory. Obviously, absolute unemployment cannot be negative as it is just a number of unemployed divided by a number of potential workers at the end of the day. Yet, according to Friedman and Phelps, there is some natural rate of unemployment ($$u^*$$). Therefore, it would be handy to break absolute unemployment into two:

$$u = u^* + u_{cycle}$$

So:

• When $$u_{cycle} = 0$$, the economy is working at its best.
• $$u_{cycle} > 0$$ is pretty much the standard situation.
• $$u_{cycle} < 0$$ is what we are interested in.

I tried to explain the existence of negative $$u_{cycle}$$ to myself the following way (and I mean from the economic perspective, not mathematical):

Say, $$f(L)$$ is our production function (having all input factors but labour constant to make things easier). We know the law of diminishing marginal productivity. Formally, we can have a situation when $$\displaystyle\lim_{L \to \inf} MP \ge 0$$. And that is the main blank point in my argument. If we agree that $$MP$$ might turn negative, it would mean that we passed through $$f$$'s optimum to the right, having the economy "overheat", not being able to perform at its best due to too many workers (which need to be paid wages).

Is my understanding right? If so, how can we be sure $$MP$$ turns negative?

• Some business cycle theories are different. E.g. imagine (a) growth phase with increasing employment and investment leading to (b) excess production with high employment leading to increasing stockpiles leading to (c) recession phase with decreasing employment and investment leading to (d) insufficient production with low employment and decreasing stockpiles leading to (a). In phase (d) you might argue $u_{cycle} > 0$ so in phase (b) you might argue $u_{cycle} < 0$. Phase (b) is not about negative marginal production, but the inability to sell marginal production at its marginal cost – Henry Aug 9 '20 at 22:46