1. You want to use it to explain incomes. But you end up having to make up all sorts of alternative types of capital in order to do so.
2. It makes it seem as if you found a planet identical to earth and inhabited only by hunter-gatherers, that you could give them all the equipment that we use to make a brand-new Honda, combine this K with their L, and produce said Honda.
3. In general, it completely overlooks the path-dependency and context-dependency of the value of capital and of labor skills. Fischer Black’s theory of economic fluctuations is focused on the fact that people make investments in an uncertain environment, and sometimes a lot of those investments turn out sour. The way I would put it is that people start down paths expecting them to lead one way, and then the paths do not lead where they want.
In general, it might be better to think of production “paths” rather than production functions. The path enabling the hunter-gatherers to build a Honda would be long and complex. It would include general-purpose technologies, like language, writing, electric motors, the internal combustion engine, computer chips, and radio-frequency communication. It would include the accumulation of human skills, know-how, infrastructure of various kinds, business norms, and regulations.
For (2) and (3), you could make a Hayekian interpretation of the production function. K measures some stock of capital, but this measure is subjective. In a very undeveloped society, automobiles are so far down their list of ranked ends that the specific capital requires to make automobiles has a very low value imputed to it. The non-intuitive result is that capital stocks — that we would consider primitive — have a higher value in very undeveloped societies than if we replaced that stock with various specific capital inputs that we use today.
Your paths get you queuing theory, which is my unproven speculation.
Perhaps an improvement on the basic production function – reflecting your observations here – would be:
Y = f(K,(L*k))
where: k = knowledge
or using your terms …
where: k = “accumulation of human skills, know-how”
or better still Y = f(K) where K = all inputs to productions